%refs.all Predrag's combined nonlinear bibliography Jun 14 2006 %mal\_refs.tex predrag 22/8-89 %mal\_ref1.tex Nonlinearity final version feb 8, 1990 %mal\_ref2.tex Nonlinearity final version feb 8, 1990 %ACK_refs.tex Artuso,C. and Kenny PR paper %Erik's thesis references %lec\_refs.tex predrag 16/6-89 %b\_refs.tex predrag 22/8-89 %v\_refs.tex predrag 22/8-89 %f\_refs.tex April 10 1990 %fieg.tex Zoltan Kovacs, sept 1991 %hhrugh.refs (thesis) july 1992 %casati.refs (book contribution-pinballs) oct 1992 %Baladi-Young refs Dec 92 %ronnie@goshawk.lanl.gov (Ronnie Mainieri) 30 Nov 92 %T. Szeredi and D.A. Goodings - Wedge Billiard - dec 92 %Braids - T.D.H. Hall, N.B. Tufillaro, 21 April 1993 %GOZZI%VXCERN@nbivax.nbi.dk Thu May 20 1993 \newcommand{\AP}[1]{{\em Ann.\ Phys. (NY)}\/ {\bf #1}} \newcommand{\CMP}[1]{{\em Commun.\ Math.\ Phys.}\/ {\bf #1}} \newcommand{\JCP}[1]{{\em J.\ Chem.\ Phys.}\/ {\bf #1}} \newcommand{\JETP}[1]{{\em Sov.\ Phys.\ JETP}\/ {\bf #1}} \newcommand{\JETPL}[1]{{\em JETP Lett.\ }\/ {\bf #1}} \newcommand{\JMP}[1]{{\em J.\ Math.\ Phys.}\/ {\bf #1}} \newcommand{\JMPA}[1]{{\em J.\ Math.\ Pure Appl.}\/ {\bf #1}} \newcommand{\JPA}[1]{{\em J.\ Phys.}\/ {\bf A #1}} %\newcommand{\JPA}[1]{{\em J.\ Phys. A: Math. Gen. }\/ {\bf #1}} \newcommand{\JPB}[1]{{\em J.\ Phys. B: At. Mol. Opt. }\/ {\bf #1}} \newcommand{\JPC}[1]{{\em J.\ Phys.\ Chem.}\/ {\bf #1}} \newcommand{\PLA}[1]{{\em Phys.\ Lett.}\/ {\bf A #1}} \newcommand{\PRA}[1]{{\em Phys.\ Rev.}\/ {\bf A #1}} \newcommand{\PRL}[1]{{\em Phys.\ Rev.\ Lett.}\/ {\bf #1}} \newcommand{\PST}[1]{{\em Phys.\ Scripta }\/ {\bf T #1}} \newcommand{\RMS}[1]{{\em Russ.\ Math.\ Surv.}\/ {\bf #1}} \newcommand{\USSR}[1]{{\em Math.\ USSR.\ Sb.}\/ {\bf #1}} % % T. Tel Jan 1990 "Transient Chaos" bibliography abreviations \def\PA#1#2{Physica~{\bf#1#2}~{\bf A}} \def\PD#1#2{Physica~{\bf D}~{\bf#1#2}} \def\PLAA#1#2{Phys.~Lett.~{\bf#1#2}~{\bf A}} \def\PLA#1#2#3{Phys.~Lett.~{\bf#1#2#3}~{\bf A}} \def\JPA#1#2{J.~Phys.~{\bf A}~{\bf#1#2}} \def\JPC#1#2{J.~Phys.~{\bf C}~{\bf#1#2}} \def\PRA#1#2{Phys.Rev.~{\bf A}~{\bf#1#2}} \def\PRB#1#2{Phys.Rev.~{\bf B}~{\bf#1#2}} \def\ZPB#1#2{Z.~Phys.~{\bf B}~{\bf#1#2}} \def\PRL#1#2{Phys.~Rev.~Lett.~{\bf#1#2}} \def\JSP#1#2{J.~Stat.~Phys.~{\bf#1#2}} \def\PTP#1#2{Prog.~Theor.~Phys.~{\bf#1#2}} \def\RMP#1#2{Rev.~Mod.~Phys.~{\bf#1#2}} \def\JP#1#2{J.~Phys.~(Paris)~{\bf#1#2}} \def\CMP#1#2{Comm.~Math.~Phys.~{\bf#1#2}} \def\ETD#1{Ergod.~Theor.~Dyn.~Syst.~{\bf#1}} \def\TAM#1#2#3{Trans.~Am.~Math.~Soc.~{\bf#1#2#3}} \def\PNA#1#2{Proc.~Natl.~Acad.~Sci.~USA~{\bf#1#2}} %%%%%%%%%%%%%%%%%%%%%% TEXTBOOKS %%%%%%%%%%%%%%%%%% \bibitem{poincare} H. Poincar\'e, {\em Les m\'ethodes nouvelles de la m\'echanique c\'eleste} (Guthier-Villars, Paris 1892-99). \bibitem{JBG97} For a very readable exposition of Poincar\'e's work and the development of the dynamical systems theory up to 1920's see J. Barrow-Green, {\em Poincar\'e and the Tree Body Problem}, (Amer. Math. Soc., Providence R.I., 1997). \bibitem{poincare-psych} H. Poincar\'e, {\em Foundations of Science}, translated by George Bruce Halsted. \bibitem{ham} R.S. MacKay and J.D. Meiss, {\em Hamiltonian Dynamical Systems} (Adam Hilger, Bristol 1987). \bibitem{almeida} A.M. Ozorio de Almeida, {\em Hamiltonian Systems: Chaos and Quantization} (Cambridge University Press, New York 1988). (The original articles are collected in two reprint collections; Hao Bai-Lin, "Chaos" (World Scientific, Singapore, 1984) and P. Cvitanovic', "Universality in Chaos" (Adam Hilger, Bristol, 1984).) \bibitem{hao} Bai-Lin Hao, {\em Chaos} (World Scientific, Singapore, 1984) \bibitem{XH94} Fa-geng Xie and Bai-lin Hao, % ``Counting the number of periods in one-dimensional maps % with multiple critical points" {\em Physica A}, {\bf 202}, 237 (1994). \bibitem{GH} %see for example J. Guckenheimer and P.J. Holmes, {\em Non-linear Oscillations, Dynamical Systems and Bifurcations of Vector Fields} (Springer, New York, 1986). \bibitem{HLB96} P. Holmes, J.L. Lumley and G. Berkooz, {\em Turbulence, Coherent Structures, Dynamical Systems and Symmetry} (Cambridge U. Press, Cambridge 1996). \bibitem{deva92} R.L. Devaney, {\em A First Course in Chaotic Dynamical Systems} (Addison-Wesley, Reading MA, 1992). \bibitem{deva87} R.L. Devaney, {\em An Introduction to Chaotic Dynamical Systems} (Addison-Wesley, Reading MA, 1987). \bibitem{CRob94} Clark Robinson, {\em Dynamical Systems: Stability, Symbolic Dynamics, \& Chaos} (C. R. C. Press, 1994) % List price US$104.95; ISBN 0849384931 \bibitem{sch} H.G. Schuster, {\em Deterministic Chaos} (need to include it because he sent me a free copy) \bibitem{BenOrsz} C.M. Bender and S.A. Orszag, {\em Advanced Mathematical Methods for Scientists and Engineers} (McGraw-Hill, New York 1978). % Chapter 11: Multiple-scale analysis \bibitem{arnold73} V.I.~Arnold, {\em Ordinary Differential Equations} (MIT Press, Cambridge, Mass. 1978) \bibitem{arnold78} V.I.~Arnold, {\em Mathematical Methods in Classical Mechanics} (Springer-Verlag, 1978, Berlin). \bibitem{arno} V.I. Arnold, {\em Geometrical Methods in the Theory of Ordinary Differential Equations} (Springer, New York 1983) \bibitem{arn1} V.I. Arnold, {\em Izv. Akad. Nauk. SSSR Math. Ser. \bf 25\rm, 21 (1961)} [{\em Am. Math. Soc. Trans. \bf 46\rm, 213 (1965)}] %cat map introduced here (says E.A. Jackson) \bibitem{AA} V.I. Arnold and A. Avez, {\em Ergodic Problems of Classical Mechanics} (Benjamin, New York 1968). %the Jacobi matrix evaluated at $t=\tau is called %"monodromy matrix" of this loop.}. V. Arnold and A.B. Givental, {\em Dynamical systems IV}, (Springer-Verlag, New York, 1990) \bibitem{Moser} J. Moser, {\em Stable and Random Motions in Dynamical Systems}, Princeton University Press, Princeton 1973 % nonzero entropy: \bibitem{Korn} I.P. Cornfeld, S.V. Fomin and Ya.G. Sinai, {\em Ergodic Theory} (Springer-Verlag, New York, 1982). \bibitem{Sinai94} Ya.G. Sinai, {\em Topics in ergodic theory} (Princeton University Press, 1994). \bibitem{Walt82} P. Walters, {\em An introduction to ergodic theory} Springer Graduate Texts in Math. Vol {\bf 79}, (Springer, New York, 1982). \bibitem{DGS76} M. Denker, C. Grillenberger and K. Sigmund, {\em Ergodic theory on compact spaces}, Springer Lecture Notes in Math., {\bf 470}, (1975). R.J.Rivers,"{\it Path-integral methods in quantum field theory.}"(Cambridge University Press, Cambridge, England, 1987). \bibitem{ll} A. J. Lichtenberg and M. A. Libermann, {\em Regular and Stochastic Motion} (Springer, New York, 1983). \bibitem{BM2} M. L. Mehta, {\it Matrix Theory: Selected Topics and Useful Results} (Les Editions de Physique, Les Ulis, 1988), pp. 107-109. See also C. Itzykson and J. B. Zuber, J. Math. Phys. {\bf 21}, 411 (1980). \bibitem{mehta90} M. L. Mehta, {\sl Random Matrix Theory}, Springer--Verlag, New York, 1990. \bibitem{Mehta} M. L. Mehta, Random Matrices, (Academic Press, New York, 1991) %Gauss measure: \bibitem{gauss} %See for example P. Billingsley, {\em Ergodic Theory and Information}, (Willey, New York 1965). \bibitem{xii. )} D. Knuth, {\em The Art of Computer Programming, Vol. II: Seminumerical Algorithms}, (Addison Wesley, Reading MA ??). \bibitem{mand} B.B. Mandelbrot, {\em The Fractal Geometry of Nature} (Freeman, San Francisco, 1983). \bibitem{Falc} %See for example K.M. Falconer, {\em The Geometry of Fractal Sets} (Cambridge Univ. Press, Cambridge, 1985) \bibitem{Edga93} G.A. Edgar, ed., {\em Classics on Fractals} (Addison Wesley, Reading MA 1993) \bibitem{sal} A. Salomaa, {\em Formal Languages} (Academic Press, San Diego, 1973). \bibitem{hop} J.E. Hopcroft and J.D. Ullman, {\em Introduction to Automata Theory, Languages, and Computation} (Addison-Wesley, Reading MA, 1979). \bibitem{cvetko} D.M. Cvetkovi\'c, M. Doob and H. Sachs, {\em Spectra of Graphs} (Academic Press, New York, 1980). \bibitem{Gelfand61} I. M. Gel'fand, {\em Lectures on Linear Algebra} (Dover, New York, 1961). % ISBN: 0486660826 \bibitem{Lang71} S. Lang, % Lang, Serge {\em Linear algebra} (Addison-Wesley, Reading, Mass. 1971). % 400 p. % QA184 .L38 1971 \bibitem{Nomizu79} K. Nomizu, % Nomizu, Katsumi {\em Fundamentals of linear algebra} (Chelsea Pub., New York 1979). % 325 p. % QA184 .N65 \bibitem{hamer} M. Hamermesh, {\em Group Theory and its Application to Physical Problems} (Addison-Wesley, Reading, 1962). \bibitem{ElBazCast72} E. ElBaz and B. Castel. {\em Graphical Methods of Spin Algebras} (M. Dekker, New York 1972) \bibitem{[12]} R. Nevanlinna, "Analytic Function", Springer - Verlag (1970). \bibitem{[8]} R. Courrant and D. Hilbert, ``Methods of Mathematical Physics", John Wiley (1989). \bibitem{Press86} W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, {\it Numerical Recipes} (Cambridge University Press, 1986), p.269. \bibitem{Luk70} Eugene Lukacs {\em Characteristic Functions} (Hafner, New York 1970). %recommended by Klauder, spring 1993 \bibitem{Saks} S.~Saks and A.~Zygmund, {\em Analytic Functions}, Elsevier, Amsterdam (1971). \bibitem{Abramowitz} M. Abramowitz and I.A. Stegun, {\em Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables}, (Dover, New York, 1964). \bibitem{edwa} H.M. Edwards, {\em Riemann's Zeta Function } (Academic, New York 1974) \bibitem{titc} E.C. Titchmarsh, {\em The Theory of Riemann Zeta Function } (Oxford Univ. Press, Oxford 1951); chapter XIV. \bibitem{encyclopedia} {\sl Encyclopedic Dictionary of Mathematics}, ed by Kiyosi It\^{o}, vol. III, MIT Press, Cambridge, 1987, 1694--1720. \bibitem{bateman} {\sl Higher Transcendental Functions}, Bateman Manuscript Project, ed. by A. Erd\'{e}lyi, vol. III, Mc. Graw--Hill, New York, 1955, 193--196. \bibitem{IP} Ivars Peterson, {\em Newton's Clock: Chaos in the Solar System} (W.H Freeman, New York 1993). Max Born, The Mechanics of the Atom (F. Ungar Publishing Co., New York 1927). \bibitem{27} K. Yosida, {\em Functional Analysis (Sixth Edition) (Springer-Verlag (Grundlehren der mathematischen Wissenschaften 123) , New York, 1980) % Floquet theory J.Cronin, {\it "Differential equations"}, Marcel Dekker, New York, 1980 % the mid point rule for the discretization. B.Sakita, {\it "Quantum theory of many variable systems and fields"}, World Scientific Publ., Singapore, 1985} \bibitem{Feynman_sp} Feynman, R.P. {\em Statistical Physics}, (Addison Wesley, New York 1990). \bibitem{UFM63} G.E. Uhlenbeck, G.W. Ford and E.W. Montroll, {\em Lectures in Statistical Mechanics} (Amer. Math. Soc., Providence R.I., 1963). \bibitem{Harris} S. Harris, {\em An introduction to the Theory of the Boltzmann Equation} (Holt, Rinehart and Winston, New York, 1971) \bibitem{Kac46} M. Kac, ``Random walk and the theory of Brownian motion'', (1946), reprinted in ref.~\cite{Wax}. \bibitem{Wax} N. Wax, ed. {\em Selected Papers on Noise and Stochastic Processes} (Dover, New York 1954). \bibitem{CJThomp72} C.J. Thompson, {\em Mathematical Statistical Mechanics}, (Macmillan, New York 1972) \bibitem{chaikin} P.M. Chaikin and T.C. Lubensky, {\em Principles of condensed matter physics}, (Cambridge University Press, Cambridge 1995). \bibitem{NegeleOr} J.W. Negele and H. Orland, {\em Quantum Many-Particle Systems} (Addison-Wesley, New York 1988) \bibitem{FH65} R.P. Feynman and A.R. Hibbs, {\em Quantum Mechanics and Path Integrals} (McGraw-Hill, New York 1965). % standard path-integral. \bibitem {Schu81} L.S. Schulman, {\em Techniques and Applications of Path Integration} (Wiley, New York, 1981). \bibitem{Griff94} D. J. Griffiths, {\em Introduction to Quantum Mechanics} (Prentice-Hall, Englewood Cliffs, New Jersey, 1994). \bibitem{Peskin95} M.E. Peskin and D.V. Schoeder, {\em An Introduction to Quantum Field Theory} (Addison Wesley, Reading MA, 1995). \bibitem{Brown92} L.S. Brown, {\em Quantum Field Theory} (Cambridge University Press, Cambridge 1992). \bibitem{Greiner} W. Greiner and J. Reinhardt, {\em Field Quantization} (Springer-Verlag, Berlin 1996). \bibitem{nelson85} Nelson E, {\em Quantum Fluctuations} (Princeton Univ. Press 1985) \bibitem{VeltmanDiagr} M. Veltman, {\em Diagrammatica: The Path to Feynman Diagrams} (Cambridge U. Press, 1994) \bibitem{DR94} W. Dittrich and M. Reuter, {\em Classical and Quantum Dynamics: From Classical Paths to Path Integrals} (Springer-Verlag, Berlin 1994) \bibitem{BH} H.P. Baltes and E.R. Hilf, {\em Spectra of finite systems} BI Wissenschaftsverlag, Mannheim 1976 \bibitem{kepler} J. Kepler, {\em Harmonices Mundi}, (Linz, 1619) % citat oversat af Helge Kragh. \bibitem{} Courant, R. and Hilbert, D. 1953. Methods of Mathematical Physics, Volume 1 (Interscience publishers); \bibitem{} Ottino, J.M., ``The kinematics of mixing: stretching, chaos and transport'' (Cambridge, 1989). \bibitem{BS} Chr. Beck, F. Schl\"ogl, {\em Thermodynamics of Chaotic Systems} (Cambridge, 1993) \bibitem{ott} E. Ott, {\em Chaos in Dynamical Systems} (Cambridge, 1993) H.-J. St\"ockmann {\em Quantum Chaos; An Introduction} (Cambridge U. Press, 1999) Kantz and Schreiber Nonlinear Time Series Analysis (Cambridge Univ. Press, Cambridge, 1997) Badii & Politi Complexity (Cambridge Univ. Press, Cambridge, 1997) J. Palis and F. Takens, Hyperbolicity & sensitive chaotic dynamics at homoclinic bifurcations (Cambridge Univ. Press, Cambridge, 1995) M.I. Vishik Asymptotic Behaviour of Solutions of Evolutionary Equations (Cambridge Univ. Press, Cambridge, 1992) \bibitem{PJD} Ph.J. Davis, {\em Circulant Matrices} (Wiley, New York, 1979) \bibitem{swl} M. Mareschal, B.L. Holian eds., {\em Microscopic Simulations of Complex Hydrodynamic Phenomena} (Plenum, New York, 1992) \bibitem{wig}S. Wiggins, {\em Chaotic Transport in Dynamical Systems} (Springer-Verlag, New York, 1992) \bibitem{vk}N.G.v. Kampen, {\em Stochastic Processes in Physics and Chemistry} (North Holland, Amsterdam, 1981) \bibitem{OSY} E. Ott, T. Sauer and J.A. Yorke, {\em Coping with Chaos} (Willey, New York 1994). \bibitem{MR94} J.E. Marsden and T.S. Ratiu, {\em Introduction to Mechanics and Symmetry} (Springer-Verlag, New York, 1994) % Section 7.10: The classical limit and the Maslov Index % contains the WKB method much like Vattay's \bibitem{MH94} J.E. Marsden and T.J.R. Hughes, {\em Mathematical Foundations of Elasticity} (Prentice-Hall, Englewood Cliffs, New Jersey, 1983) L.M. Brekhovskikh, ``Waves in layered media'' (Academic Press, New York, 1960). L.M. Brekhovskikh and O.A. Godin, ``Acoustics of layered media I: plane and quasi-plane waves'' (Springer-Verlag, Berlin 1998). % Location: SCIENCE ENGINEERING % Call number: 1 Copy Ordered as of 2/24/00 W.M. Ewing, W.S. Jardetzky and F. Press, ``Elastic waves in layered media'' (New York, McGraw-Hill, 1957). V.B. Poruchikov, ``Methods of the classical theory of elastodynamics'' (Springer-Verlag, Berlin 1993). % Location: SCIENCE ENGINEERING % Call number: 531.382 P853mX \bibitem{Kline72} M. Kline, {\em Mathematical Thought from Ancient to Modern Times} (Oxford Univ. Press, Oxford 1972); on Monge and theory of characteristics - chapter 22.7. \bibitem{LLmech} L.D. Landau and E.M. Lifshitz, {\em Mechanics} (Pergamon, London 1959). \bibitem{LLQM} L.D. Landau and E.M. Lifshitz, {\em Quantum Mechanics (nonrelativistic)} (Pergamon, London 1959) \bibitem{LifPit} E.M. Lifshitz and L.P. Pitaevskii, {\em Physical Kinetics} (Pergamon, London 1981). \bibitem{Haake91} F. Haake, {\em Quantum Signatures of Chaos} (Springer-Verlag, New York, 1991). \bibitem{BrackBh97} M. Brack and R.K. Bhaduri, {\em Semiclassical Physics} (Addison-Wesley, New York 1997). \bibitem{KJ94} H.J. Korsch and H.-J. Jodl, {\em Chaos. A Program Collection for the PC} (Springer, New York 1994) \bibitem{GMS94} M. Goossens, F. Mittelbach and A. Samarin, {\em The LaTeX Companion} (Addison-Wesley, New York 1994). Jim Meiss: I like Barger & Ollson (spelling?) undergrad mechanics text best --though it has no modern dynamics, it really kills Lagrangian and Hamiltonian systems and has nice examples --the boomerang, a grand tour of the planets, etc. %Peres: useful to anyone who wants to understand the use %of quantum theory for the description of physical processes. %graduate level text, ideal for independent study. Asher Peres, QUANTUM THEORY: CONCEPTS AND METHODS \bibitem{Dingle} R.B. Dingle {\em Asymptotic Expansions: their Derivation and Interpretation} (Academic Press, London, 1973). N. Dunford and J. Schwartz, `` Linear Operators, Part II'', Sections XI, 6,9,10, Wiley 1963 M.S. Birman and M.Z. Solomjak {\em Spectral Theory of Selfadjoint Operators in Hilbert Space} Dordrecht, Reidel (1987) H.M. Nussenzveig, {\em Diffraction effects in semiclassical scattering} Cambridge University Press 1992 \bibitem{thi} W. Thirring, ``Quantummechanics of Atoms and Molecules'', Vol. 3, (Springer, 1979) \bibitem{Rayleigh} J.W.S. Rayleigh, {\em The Theory of Sound} (Macmillan, London 1896; reprinted by Dover, New York 1945). \bibitem{Leibniz} G. W. Leibniz, {\em Von dem Verh\"angnisse}. \bibitem{Strogatz} S.H. Strogatz, {\em Nonlinear Dynamics and Chaos} (Addison-Wesley 1994). \bibitem{KatHass} A. Katok and B. Hasselblatt, {\em Introduction to the Modern Theory of Dynamical Systems} (Cambridge U. Press, Cambridge 1995). \bibitem{BH86} N. Bleistein and R.A. Handelsman, {\em Asymptotic Expansions of Integrals} (Dover, New York 1986). \bibitem{ASY96} K.T. Alligood, T.D. Sauer and J.A. Yorke, {\em Chaos, an Introduction to Dynamical Systems} (Springer, New York 1996) \bibitem{thur} W. Thurston, {\em ``On the geometry and dynamics of diffeomorphisms of surfaces''}, {\em Bull. Amer. Math. Soc. (N.S.) \bf 19}, 417 (1988). % -431. \bibitem{Bri96} J. Bricmont, {\em ``Science of Chaos or Chaos in Science?''}, available on www.ma.utexas.edu/mp_arc, \#96-116. \bibitem{tuf91} N.B.\ Tufillaro, T.A.\ Abbott, and J.P.\ Reilly, {\em Experimental Approach to Nonlinear Dynamics and Chaos} % ( Book/Disk). (Hardback. ISBN:0-201-55441-0) $36.95 (Addison Wesley, Reading MA, 1992). \bibitem{Gilm97} R. Gilmore, ``Topological analysis of chaotic dynamical systems'', {\em Rev. Mod. Phys.} (1997). \bibitem{SNM96} H. Solari, M. Natiello and G.B. Mindlin, {\em ``Nonlinear Physics and its Mathematical Tools''}, (IOP Publishing Ltd., Bristol, 1996). Safran: surfaces and membranes (Dutta recommends) John Carey, ed. The Faber Book of Science (John and Haber 1995) recommended by Michael Berry @Book{gleick, author = {Gleick, James}, title = {Chaos: Making a New Science}, publisher = {Penguin USA}, year = 1988, address = {New York, NY} } R.L. Graham, D.E. Knuth and O. Patashnik, {\em Concrete Mathematics; a Fundation for Computer Science} (Addison-Wesley, New York 1989) C. Itzykson and J.-M. Drouffe Statistical field theory, vol 1 and 2 (Cambridge U. Press, 1991) A. Knauf and Y.G. Sinai Classical Nonintegrability, Quantum Chaos 1997 approx 104pp, Birkhouser http://www.birkhauser.com. Florin Diacu & Philip Holmes Celestial Mechanics. The origin of Chaos and Stability Princeton University Press Vinay Ambegaokar Reasoning about luck; probability and its uses in physics (Cambridge Univ. Press, Cambridge, UK 1996) \bibitem{16Baladi} H. Kunita, {\em Stochastic Flows and Stochastic Differential Equations} (Cambridge University Press, Cambridge 1990) \bibitem{17 } R. Ma\~n\'e, {\em Ergodic Theory and Differentiable Dynamics (Springer-Verlag , New York 1987) %%%%%%%%%%%%%%%%%%%%%% TEXTBOOKS FINISHED %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% PREDRAG CVITANOVIC references %%%%%%%%%%%%%% \bibitem{FieldThe} P. Cvitanovi\'c, % notes prepared by Ejnar Gyldenkerne {\em Field theory} (Nordita, Copenhagen, 1983); %small parts available on {\tt www.nbi.dk/{$\sim$}predrag/field\_the/}. \bibitem{lattFT} P.~Cvitanovi\'c, {\em Lattice Field Theory}, lecture notes (Northwestern Univ., Evanston, May 1999); available on {\tt www.nbi.dk/{$\sim$}predrag/field\_the/lattFT/lattFT.ps}, \bibitem{GroupThe} %\bibitem{excep_book} P.~Cvitanovi\'c, {\em Group theory}, % part II: exceptional Lie groups}, {\tt www.nbi.dk/{$\sim$}cats/predrag/group\_the/}, a monograph in preparation %numerical and experimental attempts \bibitem{auerbach} D. Auerbach, P. Cvitanovi\'c, J.-P. Eckmann, G.H. Gunaratne and I. Procaccia, {\em Phys. Rev. Lett. \bf 58}, 2387 (1987). \bibitem{u_in_c} {\em Universality in Chaos, 2. edition}, P. Cvitanovi\'c, ed., (Adam Hilger, Bristol 1989). \bibitem{cycprl} P. Cvitanovi\'c, % Invariant measurement of strange sets in terms of cycles, {\em Phys. Rev. Lett. \bf 61}, 2729 %-2733 (1988). \bibitem{scherer} P. Scherer, {\em Quantenzust\"{a}nde eines klassisch-chaotischen Billards\/}, Ph.D. thesis, KFA J\"{u}lich, % Germany, J\"{u}l-2554, ISSN 0366-0885, (Nov. 1991). \bibitem{CERRS} P. Cvitanovi\'c, B. Eckhardt, P.E.~Rosenqvist, G.~Russberg and P. Scherer, % "Pinball Scattering", pp. ~405-433, in G. Casati and B. Chirikov, eds., {\em Quantum Chaos}, (Cambridge University Press, Cambridge 1994). \bibitem{brasil} P. Cvitanovi\'c, in {\em Nonlinear Physical Phenomena, Brasilia 1989 Winter School}, {\`A}. Ferraz, F. Oliveira and R. Osorio, eds. (World Scientific, Singapore 1990). \bibitem{boyd} P. Cvitanovi\'c, % Universal scaling laws for maps on the interval and circle maps, in R. W. Boyd, L. M. Narducci and M. G. Raymer, eds., {\em Instabilities and Dynamics of Lasers and Nonlinear Optical Systems } (U. of Cambridge Press, Cambridge, 1985). \bibitem{CJKP} P. Cvitanovi\'c, M.H. Jensen, L.P. Kadanoff and I. Procaccia, {\em``Renormalization, unstable manifolds and the fractal structure of mode locking}, {\em Phys. Rev. Lett.} {\bf 55}, 343 (1985). \bibitem{piet} P. Cvitanovi\'c, M.H. Jensen, L.P. Kadanoff and I. Procaccia, %. Circle maps in the complex plane in L. Pietronero and E. Tosatti, eds., {\em Fractals in Physics, Trieste, July 1985} (North Holland, New York, 1985). \bibitem{lund} P. Cvitanovi\'c, {\em ``Renormalization description of transitions to chaos}, in S. Lundquist, N.H. March and E. Tosatti, eds., {\em Order and Chaos in Non-linear Physical Systems}, %pp. 73-97 (Plenum, New York 1988). \bibitem{gilm} P. Cvitanovi\'c, {\em ``Hausdorff dimension of irrational windings''}, in R. Gilmore, ed., {\em Proceedings of the XV International Colloquium on Group Theoretical Methods in Physics}, %pp. 184-198 (World Scientific, Singapore, 1987). \bibitem{torino} P. Cvitanovi\'c, %4. Chaos for cyclists, in E. Moss, ed., {\em Noise and Chaos in Nonlinear Dynamical Systems,} (Cambridge Univ. Press, Cambridge 1989). \bibitem{myrh1} P. Cvitanovi\'c and J. Myrheim, % Universality for period n-tuplings in complex mappings {\em Phys. Lett. \bf 94A}, 329 (1983) \bibitem{myrh2} P. Cvitanovi\'c and J. Myrheim, {\em``Complex universality''}, {\em Commun. Math. Phys. \bf 121\rm, 225 (1989).} % this paper is in sense a continuation: \bibitem{BQT91} K.M. Briggs, GRW Quispel and C.J. Thompson, % ``Feigenvalues for Mandelsets" {\em J. Phys \bf A 24}, 3363 (1991). \bibitem{Briggs} K.M. Briggs, % ``A precise calculation of the Feigenbaum Constants'' {\em Math. of Computation}, 435 (1991) % -439 \bibitem{CSS} P. Cvitanovi\'c, B. Shraiman and B. S\"oderberg, {\em``Scaling laws for mode lockings in circle maps''}, {\em Physica Scripta \bf 32}, 263 (1985). The scaling function formalism of that reference is superceded by the cycle expansions discussed here. \bibitem{CGV} P. Cvitanovi\'c, G.H. Gunaratne and M.J. Vinson, {\em Nonlinearity \bf 3\rm, 873 (1990).} \bibitem{julia} T. Bohr, P. Cvitanovi\'c and M.H. Jensen, {\em ``Fractal aggregates in the complex plane''}, {\em Europhys. Letts. \bf 6}, 445 (1988). \bibitem{19.} The possible experimental significance of such phase transitions is discussed in P.~Cvitanovi\'c, {\em in P.Zweifel, G. Gallavotti and M.Anile, eds., \bf Non-linear Evolution and Chaotic Phenomena } (Plenum, New York 1987) \bibitem{8} P. Zweifel, G. Gallavotti and M. Anile, eds., { \em Non-linear Evolution and Chaotic Phenomena } (Plenum, New York 1987) \bibitem{phtrans} P. Cvitanovi\'c, {\em ``Phase transitions on strange sets''}, in P. Zweifel, G. Gallavotti and M. Anile, eds., {\em Non-linear Evolution and Chaotic Phenomena } (Plenum, New York 1987). % Deutsches Malloppo pinoballo: \bibitem{eck} P. Cvitanovi\'c and B. Eckhardt, {\em Periodic-Orbit Quantization of Chaotic Systems}, {\em Phys. Rev. Lett. \bf 63\rm, 823 (1989)}. \bibitem{CEflows} P. Cvitanovi\'c and B. Eckhardt, %Periodic orbit expansions for classical smooth flows {\em J. Phys. \bf A 24}, L237 (1991). \bibitem{CEsym} P. Cvitanovi\'c and B. Eckhardt, %``Symmetry decomposition of chaotic dynamics", {\em Nonlinearity \bf 6}, 277 (1993). \bibitem{CJPE} P. Cvitanovi\'c, J.-P. Eckmann and P. Gaspard, %``Transport properties of the Lorentz gas in %terms of periodic orbits", % NBI preprint (May 1991); with factorization abandoned: {\em Chaos, Solitons and Fractals \bf 6}, 113 (1995) \bibitem{ACK} R. Artuso, P. Cvitanovi\'c and B.G. Kenny, {\em ``Phase transitions on strange irrational sets''}, {\em Phys. Rev. \bf A39}, 268 (1989); P. Cvitanovi\'c, lectures in ref.~\cite{8} \bibitem{AACI} R. Artuso, E. Aurell and P. Cvitanovi\'c, % {\em ``Recycling of strange sets I: Cycle expansions", {\em Nonlinearity \bf 3}, 325 (1990). \bibitem{AACII} R. Artuso, E. Aurell and P. Cvitanovi\'c, % {\em ``Recycling of strange sets II: Applications"}, {\em Nonlinearity \bf 3}, 361 (1990). \bibitem{CCR} F. Christiansen, P. Cvitanovi\'c and H.H. Rugh, % ``The spectrum of the period-doubling operator in terms of cycles", {\em J. Phys \bf A 23}, L713 (1990). % {\em J. Phys. A : Math. Gen.} {\bf 23} (1990) 713 \bibitem{losal} P. Cvitanovi\'c, % "Periodic orbits as the skeleton of classical and quantum chaos," {\em Physica \bf D 51}, 138 (1991). \bibitem{C92} P. Cvitanovi\'c, % ``Periodic orbit theory in classical and quantum mechanics", {\em CHAOS \bf 2}, 1 (1992). % reprinted in \bibitem{gutReprints} M.C. Gutzwiller, {\em The Interplay between Classical and Quantum Mechanics} (AATPT, College Park 2001). \bibitem{CC92} P. Cvitanovi\'c and F. Christiansen, % `` Periodic orbit quantization of the anisotropic Kepler problem", {\em CHAOS \bf 2}, 61 (1992). \bibitem{CGS} P. Cvitanovi\'c, P. Gaspard, and T.~Schreiber, % ``Investigation of the Lorentz Gas in terms of periodic orbits", {\em CHAOS \bf 2}, 85 (1992). %85-90 \bibitem{C92a} P. Cvitanovi\'c, {\em Circle Maps: Irrationally Winding}, in C. Itzykson, P. Moussa and M. Waldschmidt, eds., {\em Number Theory and Physics, Les Houches 1989 Spring School}, (Springer, New York 1992). \bibitem{como90} R. Artuso, P. Cvitanovi\'c, and G. Casati, eds., {\em Chaos, Order and Patterns}, NATO Advanced Institute, Como 1990, (Plenum, New York 1992). \bibitem{EPRI} P. Cvitanovi\'c, %The Power of Chaos, in J.H. Kim and J. Stringer, eds., {\em Applied Chaos}, (John Wiley \& Sons, New York 1992). \bibitem{19.} Kvantes Lykkelige Dag, (with Kenneth Krabat) {\em Naturligvis} 20 (1991). \bibitem{qc_qm} {\em Quantum Chaos - Quantum Measurement}, P. Cvitanovi\'c, I. Percival, and A. Wirzba, eds. (Kluwer, Dordrecht, 1992). \bibitem{CHAOS92} P. Cvitanovi\'c, ed., {\em Periodic Orbit Theory - theme issue}, {\em CHAOS \bf 2}, 1-158 (1992). %Per's masters thesis \bibitem{Rosenqvist} P.E. Rosenqvist, Copenhagen University master's thesis (1991), unpublished. \bibitem{CR93} P. Cvitanovi\'c and P.E.~Rosenqvist, % ``A new determinant for quantum chaos", in G.F. Dell'Antonio, S. Fantoni and V.R. Manfredi, eds., {\em From Classical to Quantum Chaos, Soc. Italiana di Fisica Conf. Proceed. \bf 41}, pp. 57-64 (Ed. Compositori, Bologna 1993). \bibitem{CV93} P. Cvitanovi\'c and G. Vattay, {\em Entire Fredholm determinants for evaluation of semi-classical and thermodynamical spectra}, {\em Phys. Rev. Lett. \bf 71}, 4138 (1993). \bibitem{CRR93} P. Cvitanovi\'c, P.E.~Rosenqvist, H.H. Rugh, and G. Vattay, {\em A Fredholm determinant for semi-classical quantization}, {\em CHAOS \bf 3}, 619 (1993). %619-636 \bibitem{piko} P. Cvitanovi\'c, M.J.~Feigenbaum and A.S.~Pikovsky, Periodic orbit expansions for power spectra of chaotic systems, Rockefeller Univ. preprint (Oct 1993). %similar stuff in \bibitem{EG} B. Eckhardt and S. Grossmann, %``Correlation functions in chaotic systems from periodic orbits'', % Oldenburg preprint (May 1994). {\em Phys. Rev. \bf E}, Nov 1994. \bibitem{vattay_BS} G. Vattay, {\em ``Bohr Sommerfeld Quantization of Periodic Orbits"}, {\em Phys. Rev. Lett. \bf 76}, 1059 (1996). %(long version: http://www.nbi.dk/{$\sim$}predrag/QCcourse/chapter_14.ps.Z ) \bibitem{vattay_noise} G. Vattay, {\em ``Noise and quantum corrections to trace formulas''}, in \refref{QCcourse}. \bibitem{vattay_hbar} G. Vattay, {\em ``Differential equations to compute $\hbar$ corrections of the trace formula''}, (1994); {\tt chao-dyn/9406005}. \bibitem{vattay_ros} G. Vattay and P.E. Rosenqvist, {\em ``Periodic Orbit Quantization beyond Semiclassical Approximation"}, {\em Phys. Rev. Lett. \bf 76}, 335 (1996); {\tt chao-dyn/9509015}. \bibitem{BCISVdynamo} N.J.~Balmforth, P.~Cvitanovi\'c, G.R.~Ierley, E.A.~Spiegel and G.~Vattay, % ``Advection of vector fields by chaotic flows'' %(with N.J.~Balmforth, G.R. Ierley, E.A.~Spiegel and G.~Vattay), {\em Stochastic Processes in Astrophysics}, {\em Annals of New York Academy of Sciences \bf 706}, 148 (1993). \bibitem{ascona} P. Cvitanovi\'c and K.T. Hansen, %``Symbolic Dynamics and Markov Partitions for Stadium Billiard'', {\em J. Stat. Phys. }, %\bf ??}, ?? (1995). to appear. \bibitem{wedge} P. Cvitanovi\'c and K.T. Hansen, ``Symbolic dynamics of the wedge billiard", Niels Bohr Inst. preprint (Nov. 1992) \bibitem{predrag_kai} P. Cvitanovi\'c and K.T. Hansen, ``Bifurcation structures in maps of H\'enon type'', {\em Nonlinearity \bf 11}, 1233 (1998). % 1233-1261 \bibitem{averaging} P. Cvitanovi\'c, ``Dynamical averaging in terms of periodic orbits'', {\em Physica \bf D 83}, 109 (1995). \bibitem{QCcourse} P. Cvitanovi\'c, et al., {\em Classical and Quantum Chaos}, {\tt http://www.nbi.dk/ChaosBook/}, Niels Bohr Institute (Copenhagen 1999). \bibitem{cycl_book} {\em Classical and Quantum Chaos - Periodic Orbit Theory}, (with R. Artuso, R.~Mainieri, G. Vattay, et al.), {\tt http://www.nbi.dk/ChaosBook/}, advanced graduate textbook, in preparation. \bibitem{sum_rules} P. Cvitanovi\'c, Kim Hansen, J.~Rolf and G.~Vattay, {\em ``Beyond the periodic orbit theory''}, {\em Nonlinearity \bf 11}, 1209 (1998), {\tt chao-dyn/9712002}. % 1209-1232 % NBI, preprint (Sept.\ 1994). \bibitem{CCP96} F. Christiansen, P. Cvitanovi\'c and V. Putkaradze, {\em ``Hopf's last hope: spatiotemporal chaos in terms of unstable recurrent patterns''}, {\em Nonlinearity \bf 10}, 55 (1997), % 55-70 \\ {\tt chao-dyn/9606016}. \bibitem{CVW96} P. Cvitanovi\'c, G. Vattay and A. Wirzba, {\em ``Quantum fluids and classical determinants''}, in H. Friedrich and B. Eckhardt., eds., {\em Classical, Semiclassical and Quantum Dynamics in Atoms}, % -- in Memory of Dieter Wintgen}, {\em Lecture Notes in Physics \bf 485} (Springer, Heidelberg 1997) pp 29-62 \\ {\tt chao-dyn/9608012}. \bibitem{DC97} C.P. Dettmann and P. Cvitanovi\'c, ``Cycle expansions for intermittent diffusion'', %(with C.P. Dettmann), {\em Phys. Rev. \bf E 56}, 6687 (1997). \bibitem{piko1} Cycle expansion for power spectrum (with A.S.~Pikovsky), % [Conference Paper (C).] {\em Proc. SPIE - Int. Soc. Opt. Eng. (USA), \bf 2038}, 290 (1997) % pp. 290-8, 4 refs. \bibitem{noisy_Fred} P. Cvitanovi\'c, C.P.~Dettmann, R.~Mainieri and G. Vattay, {\em Trace formulas for stochastic evolution operators: Weak noise perturbation theory}, {\em J. Stat. Phys. \bf 93}, 981 (1998); \arXiv{chao-dyn/9807034}. \bibitem{conjug_Fred} P. Cvitanovi\'c, C.P.~Dettmann, R.~Mainieri and G. Vattay, {\em Trace formulas for stochastic evolution operators: Smooth conjugation method}, {\em Nonlinearity \bf 12}, 939 (1999); % 939-953. \arXiv{chao-dyn/9811003}. \bibitem{diag_Fred} P. Cvitanovi\'c, C.P.~Dettmann, G. Palla, N. S\o nderg\aa rd and G. Vattay, {\em Spectrum of stochastic evolution operators: Local matrix representation approach}, {\em Phys. Rev. \bf E 60}, 3936 (1999); % 3936-3941 \arXiv{chao-dyn/9904027}. \bibitem{CFTsketch} P. Cvitanovi\'c, {\em Chaotic Field Theory: a Sketch}, {\em Physica \bf A 288}, 61 (2000); %61-80 % (5 Nov 1999; - invited talk, Dynamics Days Asia-Pacific, % 13 - 16 July, 1999) {\tt nlin.CD/0001034} \bibitem{asym_Fred} P. Cvitanovi\'c, C.P.~Dettmann, G. Palla, N. S\o ndergaard and G. Vattay, {\em Trace formulas for stochastic evolution operators: Beyond all orders}, in preparation. % a chapter from DasBuch packaged as proceedings \bibitem{Newt97} Trace formulas in classical dynamical systems, in I.V.~Lerner, J.~Keating and D.E.~Khmelnitskii, eds., {\em Supersymmetry and Trace Formulae: Chaos and Disorder} pp. 85-102 % proceedings of a NATO ASI, Newton Institute 1997, (Plenum, New York 1998) \bibitem{tunnel} Periodic orbit theory of chaotic tunneling (with O. Sigwarth, S. Creagh and N. Whelan), in preparation \bibitem{billSumRules} Periodic orbit sum rules for billiards: Accelerating cycle expansions (with S.F. Nielsen and P. Dahlqvist), submitted to {\em J. Phys \bf A} (Jan. 1999); {\tt chao-dyn/9901001} \bibitem{linResp} Periodic orbit theory of linear response (with Niels S\o ndergaard), in preparation \bibitem{brain} M.W.~Slutzky, P.~Cvitanovic´ and D.J.~Mogul, {\em Deterministic chaos and noise in three {\em in vitro} hippocampal models of epilepsy}, %(with M.W.~Slutzky and D.J.~Mogul), {\em Annals of Biomedical Engineering \bf 29}, 607 (2001). % 607-618 %(submitted 25 may 2000) \bibitem{ratbrain} Manipulating epileptiform bursting in the rat hippocampus using chaos control and adaptive techiques (with Marc W. Slutzky and David J. Mogul), {\em IEEE Transactions on Biomedical Engineering}, to appear (submitted 29 oct 2000) \bibitem{noisybrain} Identification of determinism in noisy neuronal systems (with Marc W. Slutzky and David J. Mogul), in preparation \bibitem{sonoluminescence} Periodic orbit theory applied to a chaotically oscillating gas bubble in water (with G.~Simon, M.T.~Levinsen, I.~Csabai and \'A. Horv\'ath), {\em Nonlinearity \bf 15}, 25 (2002). %; {\tt chao-dyn/??} \bibitem{WSC01} A. Wirzba, N. Sondergaard, and P. Cvitanovic´, {\em Wave Chaos in Elastodynamic Cavity Scattering}, {\em Phys. Rev. Lett.} % (Aug 29, 2001); (2003), accepted pending revision; {\tt nlin/0108053}. \bibitem{ACT02} R.~Artuso, P.~Cvitanovi\'c and G.~Tanner, {\em Cycle expansions for intermittent maps}, submitted to {Proc. Theo. Phys. Supp.} (Dec 22, 2001); {\tt nlin/02??}. \bibitem{crete03} Y.~Lan and P.~Cvitanovi\'c, ``Turbulent fields and their recurrences,'' % (with Y.~ Lan) in N.~Antoniou, ed., {\em Proceed. of 10. Intern. Workshop on Multiparticle Production: Correlations and Fluctuations in QCD } (World Scientific, Singapore 2003); % submitted (March 2003) {\tt nlin.CD/0308006}. \bibitem{CFTsketch} {\em Chaotic field theory: a sketch}, {\em Physica \bf A 288}, 61 (2000) %61-80 % (5 Nov 1999; - invited talk, Dynamics Days Asia-Pacific, % 13 - 16 July, 1999) \\ {\tt nlin.CD/0001034} \bibitem{brain} Deterministic chaos and noise in three {\em in vitro} hippocampal models of epilepsy % M.W.~Slutzky, P.~Cvitanovic´ and D.J.~Mogul (with M.W.~Slutzky and D.J.~Mogul), {\em Annals of Biomedical Engineering \bf 29}, 607 (2001) % 607-618 %(submitted 25 may 2000) \bibitem{ratbrain} Manipulating epileptiform bursting in the rat hippocampus using chaos control and adaptive techiques (with M.W.~Slutzky and D.J.~Mogul), {\em IEEE Transactions on Biomedical Engineering}, (2003), to appear % (submitted 29 oct 2000) \bibitem{noisybrain} Identification of determinism in noisy neuronal systems (with M.W.~Slutzky and D.J.~Mogul), {\em J. Neuroscience Methods \bf 118}, 153 (2002) %153-161 % (submitted 29 Nov 2001) \bibitem{LGC02} Y.~Lan, P.~Cvitanovi\'c and N.~Garnier, ``Stationary modulated-amplitude waves in the 1-D complex Ginzburg-Landau equation.'' % (with Y.~ Lan and N.~Garnier) {\em Physica \bf D 188}, 193 (2004); % 193-212 {\tt nlin.PS/0208001}. \bibitem{ACT02} R.~Artuso, P.~Cvitanovi\'c and G.~Tanner, ``Cycle expansions for intermittent maps,'' % (with R.~Artuso, P.~Cvitanovi\'c and G.~Tanner) {Proc. Theo. Phys. Supp.} (2002), to appear; {\tt nlin.CD/0305008}. \bibitem{LanDescent} Y.~Lan and P.~Cvitanovi\'c, ``Variational method for finding periodic orbits in a general flow'', % (with Y.~Lan) {\em Phys. Rev. \bf E 69} 016217 (2004), {\tt nlin.CD/0308008} \bibitem{BECprl} M.A.~Porter and P.~Cvitanovi\'c, ``Modulated Amplitude Waves in Bose-Einstein Condensates'', %(with M.A.~Porter) {\em Phys. Rev. \bf E 69}, 047201 (2004); {\tt nonlin.CD/0307032}. \bibitem{BEC-CHAOS} M.A.~Porter and P.~Cvitanovi\'c, ``A Perturbative Analysis of Modulated Amplitude Waves in Bose-Einstein Condensates'', %(with M.A.~Porter) {\em CHAOS \bf 14}, 739 (2004); {\tt nlin.CD/0308024}. \bibitem{SondergVaxjo05} N.~Sondergaard, P.~Cvitanovi\'c, and A.~Wirzba, ´ Closed complex rays in scattering from elastic voids, % (with A.~Wirzba and N.~S\o ndergaard), in B.~Nilsson, ed., {\em Mathematical Modelling of Wave Phenomena 2005}, %August 14-19, 2005 % V\"axj\"o, Sweden (2005), {\em AIP Conference Proceedings (2006)}. %%%%%%%%%%%%%%%%%%%%%% PREDRAG CVITANOVIC references FINISHED %%%%%%%% \bibitem{PVVSD} G. Palla, G. Vattay, A Voros, N. S\o ndergaard, C.P. Dettmann, ``Noise corrections to stochastic trace formulas,'' {\em Found. Phys. \bf 31}, 641 (2001). % 641-657 \bibitem{PVV01} G. Palla, G. Vattay and A. Voros ``Trace formula for noise corrections to trace formulas,'' {\em Phys. Rev. \bf E 64}, 012104 (2001) \bibitem{Dettm03} C.P. Dettmann, ``Fractal asymptotics,'' to appear in {\em Physica \bf D} (2003). %%%%%%%%%%%%%%%%%%%%%% PRUNING FRONTS %%%%%%%%%%%%%%%%%% \bibitem{CGP} P. Cvitanovi\'c, G. H. Gunaratne and I. Procaccia, ``Topological and metric properties of H\'enon-type attractors'' {\em Phys. Rev. \bf A 38}, 1503 (1988). \bibitem{ChrPol65} F. Christiansen and A. Politi, ``A generating partition for the standard map'', {\em Phys. Rev. E. \bf 51}, 3811 (1995); {\tt \href{http://arXiv.org/abs/chao-dyn/9411005}{chao-dyn/9411005}} \bibitem{ChrPol66} F. Christiansen and A. Politi, ``Symbolic encoding in symplectic maps'', {\em Nonlinearity \bf 9}, 1623 (1996). \bibitem{ChrPol67} F. Christiansen and A. Politi, ``Guidelines for the construction of a generating partition in the standard map'', {\em Physica D \bf 109}, 32 (1997). %%%%%%%%%%%%%%%%%%%%%% PRUNING FINISHED %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% CLASSICAL DYNAMICAL SYSTEMS %%%%%%%%%%%%%% \bibitem{smale} S. Smale, %{\em Differentiable Dynamical Systems}, {\em Bull. Am. Math. Soc.} {\bf 73}, 747 (1967). \bibitem{sinai} Ya.G. Sinai, %{\em Gibbs measures in ergodic theory}, {\em Russ. Math. Surveys \bf 166}, 21 (1972). \bibitem{bowen} R. Bowen, {\em Equilibrium states and the ergodic theory of Anosov-diffeomorphisms}, Springer Lecture Notes in Math., {\bf 470}, (1975). % The Perron-Frobenius theory in the % Holder continuous case for Axiom A and % Anosov diffeos. Here, as in Ruelle 78, % the transfer op is only known to have a % "gap" and is not nuclear. In chapter 4, % the case of the weight=-log(det (F_u)), % F_u the unstable derivative, is % explained for Axiom A attractor. % (This weight corresponds to the physical measure % i.e. the time average measure - in a sense % to be made precise - and is absolutely % continuous w.r. to Lebesgue ALONG THE % UNSTABLE FOLIATION. No correl fns here % see also Ledrappier-Young Annals of Math around 86 \bibitem{Bowen1} R. Bowen, {\em Periodic orbits for hyperbolic flows}, Amer. J. Math. {\bf 94}, 1-30 (1972). \bibitem{Bowen2} R. Bowen, {\em Symbolic dynamics for hyperbolic flows}, Amer. J. Math. {\bf 95}, 429-460 (1973). \bibitem{111.} F. Ledrappier and L.-S. Young, {\em The metric entropy of Diffeomorphisms} Bull.Am.Math.Soc. {\bf 11:2}, 343-346, (1984). \bibitem{112} F. Ledrappier and L.-S. Young, Annals of Math around 86 \bibitem{birk} G.D. Birkhoff, {\em Acta Math. \bf 50}, 359 (1927), % Birkhoff variables ?? reprinted in ref. \rf{ham}. \bibitem{WPM} J. Wisdom, S. Peale and F. Mignard, % ``The chaotic rotation of Hyperion", {\em Icarus \bf 58}, 137 (1983) \bibitem{Birkhoff50} G.D. Birkhoff, {\em Collected Math. Papers}, {Vol. \bf II} (Amer. Math. Soc., Providence R.I., 1950). \bibitem{Birkhoff55} G.D. Birkhoff, {\em Dynamical systems}, Amer. Math. Soc. Colloq. Publ., {vol \bf 9}, Amer. Math. Soc., Providence R. I. (1955). \bibitem{BPV} The figures of chapter IV. of P. Berg\`e, Y. Pomeau and C. Vidal, {\em Order within Chaos}, (Wiley and Sons, New York 1984) provide a good illustration of this non-uniqueness of representations of dynamical systems. \bibitem{compinv} The hope is that this is the maximum invariant information that can be extracted from a dynamical system; however, it is not known whether cycles indeed suffice, and it is argued in ref. \rf{pres} that this is not the case. bibitem{61.} O. R\"ossler, Phys. Lett. {\bf 57A}, 397 (1976). \bibitem{LDM} C. Letellier, P. Dutertre and B. Maheu, ``Unstable periodic orbits and templates of the R\"ossler system: toward a systematic topological characterization, {\em CHAOS \bf 5}, 272 (1995). %\bibitem{64.} M. Herman, in {\em Geometry and Topology}, eds. J. Palis and M. do Carmo, {\em Lecture Notes in Math.} {\bf 597}, 271, (1977), Springer Verlag Berlin. \bibitem{65.} M. Herman, {\em Publ. IHES}, {\bf 49}, 5 (1979). (1983). \bibitem{66.} R. Thom, {\em Structural Stability and Morphogenesis}, W. A. Benjamin (1975). \bibitem{67.} S. Newhouse, in {\em Progress in Math.} {\bf 8}, Birkh\"auser (1980). \bibitem{new74} S. E. Newhouse, {\em Topology } {\bf 13}, 9 (1974) %infinity of sinks \bibitem{new79} S. E. Newhouse, {\em Publ. Math. IHES } {\bf 50}, 101 (1979) %Abundance of wild hyperbolic sets \bibitem{68.} M.Peixoto, {\em On structural stability}, Ann. of Math.(2), {\bf 69} (1959), 199-222. \bibitem{69.} M. Peixoto, {\em Structural stability on two-dimensional manifolds}, Topology {\bf 1} (1962), 101-120. % Kolmogorov-Anosov K systems, nonzero entropy: \bibitem{Anosov} D. V. Anosov, {\em Geodesic flows on closed Riemannian manifolds with negative curvature}, Proc. Steklov Inst. Math., {\bf 90}, (1967). \bibitem{anosov}D.V.Anosov, Geodezicheskiye Potoki na Zamknutych Rimanovych Mnogoobraziyach Otrizatelnoi Krivizny (Geodesic flows on closed Riemannian manifolds of negative curvature) (Nauka, Moscow, 1967) [Russian]. \bibitem{Hall93} T. Hall, ``Fat one-dimensional rpresentatives of pseudo-Anosov isotopy classes with minimal periodic orbit structure'', {\em Nonlinearity \bf 7\rm, 367 (1994).} \bibitem{east} R.W. Easton, %``Trellises formed by stable and unstable manifolds in plane {\em Trans. Am. Math. Soc.}{\bf 294}, 2 (1986). \bibitem{rom} V. Rom-Kedar, %``Transport rates of a class of two-dimensional maps and flows'' {\em Physica D} {\bf 43}, 229 (1990); %\bibitem{eule} K. Judd, % ``The fractal dimension of a homoclinic bifurcation" (1989). % ``2: Heteroclinic orbits and the Duffing system" (1989). \bibitem{kumm} M. Kummer, %On the regularization of the Kepler Problem {\em Commun. Math. Phys. \bf 84}, 133 (1982); %erratum {\bf 131}, 221 (1990). %PC: what is this paper about?: %\bibitem{llave} Llave, {\em Phys} {\bf ??}, ???? (????). % Llave says that cycles are full set of invariants for Anosovs %baker map introduced here \bibitem{hopf} E. Hopf, {\em Ergodentheorie} (Chelsea Publ. Co., New York 1948). %perhaps precursor of turbulence as reccurent patterns? \bibitem{Hopf42} E. Hopf, {\em Abzweigung einer periodischen L\"osung}, {\em Beriech. S\"achs. Acad. Wiss. Leipzig, Math. Phys. Kl. \bf 94}, 19 (1942); % 15-25. ``Bifurcation of a periodic solution from a stationary solution of a system of differential equations'', transl. by L. N. Howard and N. Kopell, in ``The Hopf bifurcation and its applications'', J. E. Marsden and M. McCracken, eds., pp. 163-193, (Springer-Verlag, New York 1976). % AMS Steele Prize give to Eberhard Hopf % for three papers of fundamental and lasting importance: % how is this different from \refref{Hopf42}? \bibitem{Hopf43} E. Hopf, ``Abzweigung einer periodischen L\"o sung von einer station\"a ren L\"o sung eines Differential systems'', {\em Beriech. S\"achs. Acad. Wiss. Leipzig, Math. Phys. Kl. \bf 95}, 3 (1943). % 3-22 %Titi says this is the precursor of turbulence as reccurent patterns? \bibitem{Hopf48} E. Hopf, A mathematical example displaying features of turbulence, {\em Commun. Appl. Math. \bf 1} (1948), 303-322. \bibitem{Hopf50} E. Hopf, The partial differential equation $u_t + uu_x = u_{xx}$, {\em Commun. Appl. Math. \bf 3} (1950), 201-230. \bibitem{[1]} R. Hide, \philtr A250, 441, 1958 \bibitem{aro_cho} D. G. Aronson, M. A. Chory, G. R. Hall and R. D. McGehee, \CMP{ 83, 303, 1982 %From KETOJA@phcu.helsinki.fi Wed Jun 16 10:57 MET 1993 \bibitem{KK93} Jukka A. 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Greenside, Comment on "Optimal periodic orbits of chaotic systems" PHYS REV LETT 80: (8) 1790-1790 FEB 23 1998 % Roman Grigoriev 13 Feb 1999 % on computation of eigenvalues/eigenvectors of chaotic systems: % K. Geist, U. Parlitz, W. Lauterborn, Prog. Theor. Phys. 83(5), p.875 (1990) % J. M. Greene, J.-S. Kim, Physica 24D, p.213 (1987) %%%%%%%%%%%%%%%%%%%%%% PERIODIC ORBITS EXTRACTION FINISHED %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% LORENZ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bibitem{lor} E.N. Lorenz, \jatm 20, 130, 1963 R. Williams % {\em The structure of Lorenz attractors}, Publ. Math. I.H.E.S., {\bf 50} (1979) 307-347, % MR\#82b:58055b \bibitem{EO93} B. Eckhardt and G. Ott, % ``Periodic orbit analysis of the Lorenz attactor", {\em Z. f. Physik \bf B 93}, 259 (1994). \bibitem{FGZ93} V. Franchescini, C. 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Politi, %``Geometric-Properties of the Pruning Front {\em PROGRESS OF THEORETICAL PHYSICS 1991, Vol 86, Iss 6, pp 1149-1157 @article {MR93a:58052, AUTHOR = {D'Alessandro, G. and Isola, S. and Politi, A.}, TITLE = {Geometric properties of the pruning front}, JOURNAL = {Progr. Theoret. Phys.}, FJOURNAL = {Progress of Theoretical Physics}, VOLUME = {86}, YEAR = {1991}, NUMBER = {6}, PAGES = {1149--1157}, ISSN = {0033-068X}, CODEN = {PTPKAV}, MRCLASS = {58F03 (58F13 58F20)}, MRNUMBER = {93a:58052}, MRREVIEWER = {Christian Beck}, } \bibitem{#} Livi, R Politi, A Ruffo, S %``Repeller Structure in a Hierarchical Model .2. Metric Properties {\em J. Stat. Phys. \bf 1991, Vol 65, Iss 1-2, pp 73-95 \bibitem{sim} S. Simo, {\em J. Stat. Phys. \bf 21}, 21 (1979). \bibitem{mis2} M. Misiurewicz, in {\em Non-linear dynamics, Annals of the New York Academy of Sciences}, Vol{\bf 357}, R. H. G. Helleman ed, (1980)p. 348. \bibitem{tel} T. Tel, {\em J. Stat. 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A solution of the pruning front conjecture and the first tangency problem'', {\em Nonlinearity} 731 (1997). @article {MR98h:58117, AUTHOR = {Ishii, Yutaka}, TITLE = {Towards a kneading theory for {L}ozi mappings. {I}. {A} solution of the pruning front conjecture and the first tangency problem}, JOURNAL = {Nonlinearity}, FJOURNAL = {Nonlinearity}, VOLUME = {10}, YEAR = {1997}, NUMBER = {3}, PAGES = {731--747}, ISSN = {0951-7715}, CODEN = {NONLE5}, MRCLASS = {58F13 (54H20 58F03)}, MRNUMBER = {98h:58117}, MRREVIEWER = {Frederick R. Marotto}, } \bibitem{KTH92} K.T. Hansen, % ``Pruning of orbits in 4-disk and hyperbola billiards", {\em CHAOS \bf 2}, 71 (1992). @article {MR93a:58053, AUTHOR = {Hansen, Kai T.}, TITLE = {Pruning of orbits in four-disk and hyperbola billiards}, JOURNAL = {Chaos}, FJOURNAL = {Chaos. An Interdisciplinary Journal of Nonlinear Science}, VOLUME = {2}, YEAR = {1992}, NUMBER = {1}, PAGES = {71--75}, ISSN = {1054-1500}, CODEN = {CHAOEH}, MRCLASS = {58F03 (58F13)}, MRNUMBER = {93a:58053}, MRREVIEWER = {Valery Covachev}, } \bibitem{Carvalho} A. de Carvalho, Ph.D. thesis, CUNY New York 1995. % Date: Tue, 26 Jun 2001 11:18:58 -0400 (EDT) % From: Andre de Carvalho @Unpublished{dCH2, author = {de Carvalho, A. and Hall, T.}, title = {The forcing relation for horseshoe braid types}, note = {IMS Preprint \#2001/02}, year = {2001} } @Unpublished{dCH2, author = {de Carvalho, A. and Hall, T.}, title = {Pruning theory and {T}hurston's classification of surface homeomorphisms}, note = {To appear in J. European Math. Soc.}, year = {2001} } @Article{dC, author = {de Carvalho, A.}, title = {Pruning fronts and the formation of horseshoes}, journal = {Ergodic Theory Dynam. Systems}, year = {1999}, volume = {19}, number = {4}, pages = {851--894} } \bibitem{Carvalho} A. de Carvalho and T. Hall, ` How to prune a horseshoe'', {\em Nonlinearity \bf 15}, R19 (2002). % pp. R19-R68 \bibitem{ted} L. Tedeschini-Lalli and J.A. Yorke, % {\em How often do simple dynamical processes % have many coexisting sinks?} {\em Commun. Math. Phys. \bf 106}, 635 (1987). \bibitem{dawson_grebogi_kocak} S.P. Dawson, C. Grebogi, and H. Ko\c{c}ak, % Geometric mechanism for antimonotonicity % in scalar maps with two critical points {\em Phys. Rev. \bf 48}, 1676 (1993). \bibitem{yorke2} C. Grebogi, E. Ott and J. Yorke, % Unstable periodic orbits and the dimension of % multifractal chaotic attractors {\em Phys. Rev. A \bf 37, \rm 1711 (1988)}. %%%%%%%%%%%%%%%%%%%%%%% HENON FINISHED %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% STADIUM %%%%%%%%%%%%%%%%%%%%%%%%%% %stadium references \bibitem{buni74} L.A. Bunimovich, % {\em Funkts. Anal. Ergo. Prilozh. \bf 8}, 73 (1974). {\em Funct. Anal. Appl. \bf 8}, 254 (1974). \bibitem{buni79} L.A. Bunimovich, {\em Comm. Math. 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Koga, ``Statistical Properties of Unstable Periodic Orbits in the stadium billiard'', (Yamanashi Medical College preprint, Sept. 1994). %%%%%%%%%%%%%%%%%%%%%% STADIUM FINISHED %%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% BILLIARDS, WEDGE BILLIARD %%%%%%%%%%%%%%%%%%%%%% \bibitem{LM86} H.E. Lehtihet and B.N. Miller, {\em Physica \bf D 21}, 93 (1986). \bibitem{MR88} B.N. Miller and K. Ravishankar, % wedge billiard Lyapunov's close to integrability {\em J. Stat. Phys. \bf 53\rm, 1299 (1988).} \bibitem{W90} M.P. Wojtkowski, %`` A system of one dimensional balls with gravity", {\em Commun. Math. Phys. \bf 126}, 507 (1990). \bibitem{W90a} M.P. Wojtkowski, %`` A system of one dimensional balls in an external field. II", {\em Commun. Math. Phys. \bf 127}, 425 (1990). \bibitem{Whelan90} N.D. Whelan, D.A. Goodings and J.K. Cannizzo, {\em Phys. Rev. \bf A 42}, 742 (1990). \bibitem{RSW90} P.H. Richter, H-J. Scholz and A. Wittek, % `A Breathing Chaos' {\em Nonlinearity \bf 1}, 45 (1990). \bibitem{Chernov91} N.I. Chernov, {\em Physica \bf D 53}, 233 (1991). \bibitem{Szeredi92} T. Szeredi, Ph.D. thesis, McMaster University (1992). \bibitem{GS91} D.A. Goodings and T. Szeredi, {\em Am. J. Phys. \bf 59}, 924 (1991). \bibitem{SG92} T. Szeredi and D.A. Goodings, {\em Phys. Rev. Lett. \bf 69}, 1640 (1992). \bibitem{SG93} T. Szeredi and D.A. Goodings, %``Classical and Quantum Chaos of the Wedge Billiard % I and II" Phys. Rev. {\bf E 1}, to appear (1993). \bibitem{LSG93} J.H. Lefebvre, T. Szeredi and D.A. Goodings, to be published. N. Berglund Billiards in a potential: variational methods, periodic orbits and KAM tori mp_arc@math.utexas.edu -96-341 - % classical motion of a particle in a plane % domain, under the influence of a perpendicular magnetic field % and a smooth potential \bibitem{GU90} M.J. Giannoni and D. Ullmo, ``Coding chaotic billiards: I. Non-ompact billiards on a negative curvature manifold'', {\em Physica \bf D 41}, 371 (1990). \bibitem{UG95} D. Ullmo and M.J. Giannoni, ``Coding chaotic billiards: II. Compact billiards defined on the psudosphere'', {\em Physica \bf D 84}, 329 (1995). \bibitem{billSumRules} S.F.~Nielsen, P.~Dahlqvist, P.~Cvitanovi{\a'{c}}, % Sune F. Nielsen, Per Dahlqvist, Predrag Cvitanovi{\a'{c}} % Periodic orbit sum rules for billiards: Accelerating cycle expansions {\em J. Phys. \bf A 32}, 6757 (1999), % 6757-6770 {\tt chao-dyn/9901001} \bibitem{chernov} N.~Chernov, %{\em Entropy, Lyapunov exponents and mean free path for billiards}, {\em J. Stat. Phys. \bf 88}, 1 (1997). \bibitem{Abramov} L.~M.~Abramov, {\em Dokl.\ Akad.\ Nauk.\ SSSR \bf 226}, 128, (1959). %%%%%%%%%%%%%%%%%%%%%% BILLIARDS FINISHED %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% FEIGENBAUM references %%%%%%%%%%%%%%%%%%%%%% \bibitem{funceq} M.J. Feigenbaum, {\em J. Stat. Phys. \bf 21\rm, 669 (1979).} \bibitem{pd} M. J. Feigenbaum, {\it J. Stat.\ Phys.\ }{\bf 19}, 25 (1978); {\bf 21}, 669 (1979). \bibitem{scal} M. J. Feigenbaum, {\it Phys.\ Lett.\ }{\bf 74A}, 375 (1979). \bibitem{15.} M.J. Feigenbaum, {\em Comm. Math. Phys. \bf 77}, 65 (1980). \bibitem{3} M.J. Feigenbaum, L.P. Kadanoff, S.J. Shenker, {\em Physica }{\bf 5D}, 370 (1982). \bibitem{markov} M.J. Feigenbaum, {\em J. Stat. Phys. } {\bf 46}, 919 (1987); {\bf 46}, 925 (1987). %the paper with golden mean pres function \bibitem{feignonlin} see for ex. M.J. Feigenbaum, {\em Nonlinearity } {\bf 1}, 577 (1988). %brutal functional iteration: \bibitem{fpt} M.J. Feigenbaum, I. Procaccia and T. T\'{e}l, {\em Phys. Rev. A} {\bf 39}, 5359 (1989). \bibitem{pres} M.J. Feigenbaum, {\em J. Stat. Phys} {\bf 52}, 527 (1988). \bibitem{7.} M.J. Feigenbaum, in ref. \rf{zweif}. \bibitem{22} M.J. Feigenbaum and R.D. Kenway, in {\em Proceedings of the Scottish Universities Summer School}, (1983); %%%%%%%%%%%%%%%%%%%%%% FEIGENBAUM references FINISHED %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% RENORMALIZATION %%%%%%%%%%%%%% \bibitem{ColEck} P. Collet and J.--P. Eckmann, {\em Iterated Maps on the Interval as Dynamical Systems} (Birkhauser, Boston, 1980). \bibitem{col} P. Collet, J.-P. Eckmann, and H. Koch, {\it J. Stat.\ Phys.\ }{\bf 25}, 1 (1981). bibitem{col} P. Collet and Y. Levy, \CMP{93}, 461 (1984) \bibitem{[14]} J. P. Eckmann, {\em Rev. Mod. Phys. \bf 53}, 643 (1981) \bibitem{24.} E.B. Vul, Ya.G. Sinai and K.M. Khanin, {\em Uspekhi Mat. Nauk \bf 39}, 3 (1984) {\em Russian Math. Surveys \bf 39}, 1 (1984). \bibitem{pdgrass} P. Grassberger, {\em J. Stat. Phys. \bf 26}, 173 (1981). Derida, Gervais and Pomeau Procaccia, Tresser and Thomae Eckmann, H. Epstein and Wittwer \bibitem{nau} M. Nauenberg and J. Rudnick, {\it Phys.\ Rev.\ }{\bf B24}, 439 (1981). \bibitem{guna2} G.H. Gunaratne, doctoral thesis (Cornell University, 1986). % per-doubling reppeller dimension: \bibitem{pdgrass} P. Grassberger, {\em J. Stat. Phys. \bf 26}, 173 (1981) \bibitem{pdaur} E. Aurell, {\em Phys. Rev. \bf A34}, 5135 (1986); {\bf A35}, 4016 (1987). \bibitem{#} Aurell, E %``Finding Eigenvalues of the Period-Doubling Operator from the Characteristic Equation {\em ANNALES DE L INSTITUT HENRI POINCARE-PHYSIQUE THEORIQUE 1990, Vol 53, Iss 4, pp 467-477 \bibitem{lib} A. Libchaber and J. Maurer, {\it J. Phys.\ (Paris) Coll.\ }{\bf 41}, C 3--51 (1980). \bibitem{sullivan} D. Sullivan, in ref.~\cite{u_in_c}. \bibitem{lanf} O. E. Lanford III, {\em Bull. Am. Math. Soc. \bf 6\rm, 427 (1982).}; reprinted in ref.~\cite{u_in_c}. \bibitem{27.} M.J.Feigenbaum, {\em J.Stat.Phys. \bf 46}, 919 (1987); {\bf 46}, 925 (1987) \bibitem{pdgrass2} P. Grassberger, {\em J. Stat. Phys. \bf 26}, 173 (1981). \bibitem{pdfalfa2circ} L.P. Kadanoff, {\em J. Stat. Phys. \bf 43}, 395 (1986). \bibitem{pdfalfa2} D. 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Frisch, {\em Fully Developed Turbulence and Intermittency}, in Proc. of Int. School on {\em Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics}, M. Ghil editor, North-Holland, (1984). \bibitem{44.} M. Jensen, L. Kadanoff, A. Libchaber, I. Procaccia and J. Stavans, %{\em Global Universality at the onset of Chaos: Results of a Forced %Rayleigh-B\'enard Experiment}, Phys. Rev. Lett., {\bf 55}, 2798, (1985). U. Frisch and G. Parisi, "Varenna School LXXXXVIII", M. Ghil, R. Benzi, and G. Parisi, eds., North-Holland, New York (1985), p.84 \bibitem{3.} B.B. Mandelbrot, {\em J. Fluid. Mech. \bf 62}, 331 (1974). \bibitem{4.} G. Paladin and A. Vulpiani, {\em Phys. Rep. \bf 156}, 147 (1987), and references therein. \bibitem{5.} X.-Z. Wu, L.P. Kadanoff, A. Libchaber and M. Sano, {\em Phys.Rev.Lett. \bf 64}, 2140 (1990). \bibitem{6.} L.P. Kadanoff, S.R. Nagel, L. Wu, and S.-m. Zhou, Phys.Rev.A \bf{39}, 6524 (1989). \bibitem{83.} A. 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Phys. \bf 46}, 919 (1987); {\bf 46}, 925 (1987). \bibitem{chicago5} T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia and B.I. Shraiman, {\em Phys. Rev. \bf A 107}, 1141 (1986). \bibitem{moron} D. Bensiomon, T.C. Halsey, M.H. Jensen, L.P. Kadanoff, A.~Libchaber, I.~Procaccia, B.I.~Shraiman and J.~Stavans, {\em``More on microcanonical paradigm''}, (G\"oteborg 17 Nov. 1986), repeatedly rejected from various proceedings. \bibitem{falfa} Even though the thermodynamic formalism is of older vintage (we refer the reader to ref.~\cite{ruelle} for a comprehensive overview), we adhere here to the notational conventions of ref.~\cite{chicago} which are more current in the physics literature. \bibitem{barnsley} M. Barnsley, {\em Fractals Everywhere} (Academic Press, New York 1988). \bibitem{shannon} C. Shannon, {\em A mathematical theory of Communication}, {\em Bell System Technical Journal, \bf 27}, 379 (1948). \bibitem{liap} H. Fujisaka, {\em Progr. Theor. Phys.} {\bf 70}, 1264 (1983). % the major paper exposing the theory of nonzero Lyapunov exponents \bibitem{pesin} Ya.B. Pesin, {\em Uspekhi Mat. Nauk \bf 32, \rm 55 (1977)}, [{\em Russian Math. Surveys \bf 32, \rm 55 (1977)}] % this paper is for (dissipative) hyperbolic attractor with singularities \item{[P]} Ya.B. Pesin, {\it Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties}, {\em Ergodic Theory and Dynamical Systems, \bf 12}, pp.123-151, 1992. \bibitem{nonhyp} A. Politi, R. Badii and P. Grassberger, {\em J. Phys. A \bf 15}, L763 (1988); {\em Scaling Laws for Invariant Measures on Hyperbolic and Nonhyperbolic Attractors} P. Grassberger, R. Badii and A. Politi, {\em J. Stat. Phys. \bf 51}, 135 (1988). %countable Markov partitions introduced here: \bibitem{hof1} F. Hofbauer, {\em ``Periodic points for piecewise monotone transformations''}, {\em Ergod. The. and Dynam Sys. \bf 5}, 237 (1985). \bibitem{hof2} F. Hofbauer, {\em ``Piecewise invertible dynamical systems"}, {\em Prob. Th. Rel. Fields \bf 72}, 359 (1986). % Markov diagrams for unimodal maps: \bibitem{grasplex} P. Grassberger, {\em Z. Naturforsch. \bf 43 a}, 671 (1988) \bibitem{MR1} D.H. Mayer and G. Roepsdorff, {\em J. Stat. Phys. \bf 47}, 149 (1987).%--171 \bibitem{MR2} D.H. Mayer and G. Roepsdorff, {\em J. Stat. Phys. \bf 50}, 331 (1987).%--344 \bibitem{mayer} D.H. Mayer, {\em Bull. Soc. Math. France} {\bf 104}, 195 (1976). % `` On a $\zeta$ function related to the % continued fraction transformation % 195--203 %On the thermodynamic Formalism for the Gauss Map: \bibitem{may90} D.H. Mayer, {\em Commun. Math. Phys. \bf 130}, 311 (1990). %311--333 \bibitem{Mayer91} D.H. Mayer, {\em Continued fractions and related transformations}, in ref.~\cite{BKS91}. \bibitem{BKS91} T. Bedford, M.S. Keane and C. 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Physics \bf 188}, 221 (2004). % 221-259 \bibitem{peyret} R.~Peyret, {\em Spectral Methods for Incompressible Viscous Flow} (Springer, Berlin 2002). %%%%%%%%%%%%%%%%%%%%% TURBULENCE FINISHED %%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% EXTENDED SYSTEMS %%%%%%%%%%%%%%%%%%%%%%%%%%% \bibitem{[10]} Y. Kuramoto, {\it Chemical Oscillations, Waves and Turbulence\/} (Springer, Berlin, 1984). \bibitem{Kur2} Kuramoto Y., Progr. Theor. Physics {\bf 71}, (1984) 1182. \bibitem{Shi} Sivashinsky G.I., Ann. Review of Fluid Mech. {\bf 15}, (1983) 179. \bibitem{KS} Y. Kuramoto and T. Tsuzuki, ``Persistent propagation of concentration waves in dissipative media far from thermal equilibrium," {\em Progr. Theor. Physics \bf 55}, 365 (1976); %365-369 G.I. Sivashinsky, ``Nonlinear analysis of hydrodynamical instability in laminar flames - I. Derivation of basic equations,'' {\em Acta Astr. \bf 4}, 1177 (1977). %1177-1206 \bi{KNS90} I.G. Kevrekidis, B. Nicolaenko and J.C. Scovel, {\em ``Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation}, {\em SIAM J. Applied Math. \bf 50}, 760 (1990). \bibitem{CF} P. Coullet and S. Fauve, {\em ``Collective modes of periodic structures''}, Combustion, flames and fires, Les Houches (1984) Eds. de Physique. I. Procaccia, T. Bohr, M. H. Jensen, V. L'vov, K. Sneppen, and R. Zeitak: ``Surface Roughening and the Long-Wavelength Properties of the Kuramoto-Shivashinsky", Phys. Rev. A, ?. \bi{HNZ} J.M. Hyman, B. Nicolaenko and S. Zaleski, \PD 23, 265 (1986) %%%%%%%%%%%%%%%%%%%%% EXTENDED SYSTEMS FINISHED %%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% NUMBER THEORY %%%%%%%%%%%%%%%%%%%%%%%%%%% \bibitem{hardy} G.M. Hardy and E.M. Wright, {\em The Theory of Numbers}, (Oxford Univ. Press, Oxford 1938). % Farey sequence: \bibitem{NZM} I. Niven, H. S. Zuckerman, and H. L. Motgomery, An Introduction to the Theory of Numbers, 5th ed., John Wiley, New York, 1991. \bibitem{KKL} A.Ya. Khinchin, {\em Continued Fractions} (U. of Chicago Press, Chicago, 1964) % Kuzmin, Levy ?? \bibitem{eule} L. Euler, {\em Opera Omnia} (Teubner, 1922). %%----------Stellan's refs on Number Theory and Ergodicity: M. Kac, eds. Baclawski and Dowder, {\em Probability, Number Theory and Statistical Physics, Selected papers} (MIT Press, 1979). M. Kac, {\em Probability, Statistical Mechanics and Number Theory} % ed Rota (Academic Press, 1986). M. Kac and S. Ulam, (Praeger, 1986). \bibitem{Kac66}M.Kac, Am. Math. Mon. {\bf 73}, 1 (1966). %Paper: hep-th/9304052 : milton@phyast.nhn.uoknor.edu (Kim Milton) %Date: Tue, 13 Apr 93 13:26:21 CDT % CONTINUED FRACTION AS A DISCRETE NONLINEAR TRANSFORM} % Carl M. Bender, Kimball A. Milton \bibitem{BM1} H. S. Wall, {\it Analytic Theory of Continued Fractions} (Van Nostrand, New York, 1948), p. 197; W. B. Jones and W. J. Thron, {\it Continued Fractions: Analytic Theory and Applications\/} (Addison-Weitey, Reading, MA, 1980), pp. 250-255. \bibitem{BM3} For the results on Euler and Bernoulli numbers see Ref. BM2, p. 323. See also H. Au-Yang and J. Perk, Physica {\bf 144A}, 44 (1987). \bibitem{BM4} C. M. Bender and W. E. Caswell, J. Math. Phys. {\bf 119}, 2579 (1978). %---------- Bender finished ---------------------------------------- Fee and Granville, Math. Comp. 57, 839 (1991) Moebius function appears as f(z)=\prod_{k=1}^\infty (1-z^k)^{\mu(k)} %%%%%%%%%%%%%%%%%%%%% NUMBER THEORY FINISHED %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% FAREY NUMBERS %%%%%%%%%%%%%%%%%%%%%%%%%%% \bibitem{CFM} R.M. Corless, G.W. Frank and J.G. Monroe % Chaos and Continued Fractions {\em Physica } {\bf D 46}, 241 (1990). \bibitem{CS} A. Csord{\'a}s and P. Sz{\'e}pfalusy, {\em Phys. Rev. }{\bf A 40}, 2221 (1989) and references therein % viviane Baladi has tracked Farey Farey intersections to % older literature - you might be interested: \bibitem{szekeres} G. Szekeres, Multidimensional continued fractions Annales Univ. Sci. Budapest, Sectio Math. 13, 113 (1970?). \bibitem{hurwitz94} A. Hurwitz, Math. Ann. 44, 417 (1894). % Vattay: about `M' or `\kappa' and its `operator-valued continued fraction' % in higher than 2 dimensions. {\em Comm. Math. Phys. \bf 129}, 535-560 (1990) %has up-to-date references on Farey Sequences and such: \bibitem{Hall_T} R.R. Hall and G. Tenenbaum, ``The set of multiples of a short interval", in D.V. Chidinovsky {em et al.}, eds., {\em Number Theory - New York Seminar 1989-1990} (Springer-Verlag , New York 1987) \bibitem{32.} R.R.Hall, {\em J.London Math.Soc., \bf 2}, 139 (1970). \bibitem{33.} S.Kanemitsu, R.Sita Rama Chandra Rao and A.Siva Rama Sarma, {\em J.Math.Soc. Japan \bf 34}, 125 (1982). \bibitem{34.} R.R. Hall and G. Tenenbaum, {\em Acta Arith. \bf 44}, 397 (1984). \bibitem{Fareytree} G.T. Williams and D.H. Browne, {\em Amer. Math. Monthly \bf 54}, 534 (1947). \bibitem{mackay} R.S. MacKay, doctoral thesis (Princeton University, 1982). \bibitem{0_mes} For the numerical evidence see refs. \cite{1,lanf}. The proof that the set of irrational windings is of zero Lebesgue measure is given in ref.~\cite{Swiatek} \bibitem{lanf} O.E. Lanford, {\em Physica }{\bf 14D}, 403 (1985). % locally maximal hyperbolic sets with periodic orbits dense % defined in \bibitem{lanf84} O.E. Lanford, Erice lectures, Dynamic Systems, Velo and Wightman eds., (1984?) \bibitem{Swiatek} G. Swiatek, %"Rational Rotation Numbers for Maps of the Circle" {\em Commun. Math. Phys.} {\bf 119}, 109 (1988). \bibitem{2} S.J. Shenker, {\em Physica }{\bf 5D}, 405 (1982). \bibitem{Lanf} O.E. Lanford, in CIME, Ciclo 1976, {\em Statistical Mechanics}, Liguori Editore, pp 25-98 \bibitem{10} O.E. Lanford, { in M. Mebkhout and R. S\'en\'eor, eds., \em Proc. 1986 IAMP Conference in Mathematical Physics } (World Scientific, Singapore 1987); lectures in ref.~\cite{8}; D. Rand, {\em Nonlinearity} {\bf 1}, 78 (1988). \bibitem{12} S-H. Kim and S. Ostlund, %Universal scaling in the circle map, {\em Physica }{\bf D 39}, 365 (1989). Phys Rev A 86 on the 2-dim Farey thing; Phys Rev Lett 85 on renormalisations of the 2-torus Physica Scripta 85 on one-dimensional things Physica D 88 which contains too much Fourier analysis \bibitem{19} K. Kaneko, {\em Prog.Theor.Phys. }{\bf 68}, 669 (1982); {\bf 69}, 403 (1983); {\bf 69}, 1427 (1983). \bibitem{FS} J.D. Farmer and I.I. Satija, {\em Phys. Rev. }{\bf A 31}, 3520 (1985). \bibitem{UFS} D.K. Umberger, J.D. Farmer and I.I. Satija, {\em Phys. Lett. }{\bf A 114}, 341 (1986). \bibitem{PM} Y. Pomeau and P. Manneville, {\em Commun. Math. Phys.} {\bf 74}, 189 (1980). \bibitem{39.} D.A.Smith, and W.F.Ford, {\em SIAM J.Numer.Anal. \bf 16}, 223 (1979). \bibitem{41.} J.M.Vanden Broeck and L.W.Schwartz, {\em SIAM J.Numer.Anal. \bf 10}, 658 (1979). \bibitem{42.} M.N.Barber, in C.Domb and J.L.Lebowitz, eds., {\em Phase Transitions and Critical Phenomena } (Academic, New York 1983), p. 226. \bibitem{20} D. Levin, {\em Inter. J. Computer Math. }{\bf B3}, 371 (1973). \bibitem{20a} Osada, siam j.numer.anal. 27 (1990) 178: a convergence acceleration method for some logarithmically convergent sequences... %includes some review on other methods (like levin.) \bibitem{17} R. Artuso, doctoral thesis, (University of Milano, 1988). \bibitem{SK} S.J. Shenker and L.P. Kadanoff, {\em J. Stat. Phys. } {\bf 27}, 631 (1982) \bibitem{JBB} For a nice discussion of physical applications of circle maps, see for example refs.~\cite{1}. \bibitem{JBB83} M.H. Jensen, P. Bak, T. Bohr, {\em Phys. Rev. Lett. \bf 50}, 1637 (1983); {\em Phys. Rev. \bf A 30}, 1960 (1984); P. Bak, T. Bohr and M.H. Jensen, {\em Physica Scripta \bf T9}, 50 (1985), reprinted in ref.~\cite{uchaos} \bibitem{ROSS} %D. Rand, S. Ostlund, J. Sethna and E. Siggia, % {\em Phys. Rev. Lett. }{\bf 49}, 132 (1982); S. Ostlund, D.A. Rand, J. Sethna and E. Siggia, {\em Physica }{\bf D 8}, 303 (1983) \bibitem{herm} M. Herman, {\em Publ. IHES}, {\bf 49}, 5 (1979). %conjugating subcritical circle map to rotation \bibitem{yocc} J.-C. Yoccoz, {\em Ann. Scient. {\'E}. norm. sup., Paris \bf 17\rm, 333 (1984)} %conjugating subcritical circle map to rotation \bibitem{glass} L. Glass, M.R. Guevara, A. Shrier and R. Perez, {\em Physica \bf D 7\rm, 89 (1983)}, reprinted in ref.~\cite{uchaos} \bibitem{21.} J.Maselko and H.L.Swinney, {\em Phys.Rev.Lett. \bf 55}, 2366 (1985) \bibitem{22.} J.Maselko and H.L.Swinney, {\em J.Chem.Phys. \bf 85}, 6430 (1986); {\em Phys.Lett. \bf A119}, 403 (1987) \bibitem{0_mes} For the numerical evidence see refs. \cite{1,lanf}. The proof that the set of irrational windings is of zero Lebesgue measure is given in ref.~\cite{swia}. \bibitem{fart} The Farey tree partitioning was introduced in refs.~{\cite{Fareytree,mackay,myrh,CSS}} and its thermodynamics is discussed in detail in refs.~{\cite{ACK,pres}}. \bibitem{pres} M.J. Feigenbaum, lectures in ref.~\cite{8}; {\em J. Stat. Phys. } {\bf 52}, 527 (1988) \bibitem{Knauf_10} A. Knauf, ``On a ferromagnetic spin chain'', {\em Commun. Math. Phys.} {\bf 153}, 77 (1993). \bibitem{Knauf_11} A. Knauf, ``Phases of the number-theoretical spin chain'', {\em J. Stat. Phys. } {\bf 73}, 423 (1993). \bibitem{Knauf_12} A. Knauf, ``On a ferromagnetic spin chain. Part II: Thermodynamic limit'', {\em J. Math. Phys.} {\bf 35}, 228 (1994). \bibitem{CK95} P. Contucci and A. Knauf, ``The phase transition of the number-theoretical spin chain'', {\em Forum Mathematicum \bf 9}, 547--567 (1997). % (T.U. Berlin preprint Sfb 288, No. 172, June 1995). \bibitem{mest} A computer-assisted proof for the golden--mean winding number has been caried out by B.D. Mestel, Ph.D. Thesis (U. of Warwick 1985). %Lanford-like proof for circle maps singel exp eigenvalue We assume that there is a single expanding eigenvalue for any periodic renormalization. \bibitem{hall} R.R. Hall, {\em J. London Math. Soc., \bf 2}, 139 (1970) \bibitem{kanemi} S. Kanemitsu, R. Sita Rama Chandra Rao and A. Siva Rama Sarma, {\em J. Math. Soc. Japan \bf 34}, 125 (1982); {\em Acta Arith. \bf 44}, 397 (1984) \bibitem{FL} J. Franel and E. Landau, {\em G\"ottinger Nachr. 198} (1924) \bibitem{nevi} E.H. Neville, {\em Roy. Soc. Mathematical Tables }(Cambridge U. Press, Cambridge 1950) \bibitem{SB} G. Schmidt and J. Bialek, {\em Physica }{\bf 5D}, ?? (1982) %Hamiltonian fractal diagram \bibitem{21} J. Stavans, F. Heslot and A. Libchaber, {\em Phys. Rev. Lett.} {\bf 55}, 569 (1985). \bibitem{GW} E.G. Gwinn and R.M. Westervelt, {\em Phys. Rev. Lett. \bf 59}, 157 (1987) % From: cycler@mynah.LANL.GOV 24 Feb 1993 Paper: 93feb007 % Title: CYCLES AND CIRCLES IN ROUNDOFF ERRORS % Author: George G. Szpiro NZZJRS@DM.RS.CH % When a series of measurements is performed with increasingly coarse % (or increasingly fine) precision, consecutive observations seem to % be erratically distributed at first, and then organize themselves % into cycles and patterns. The patterns, which arise because of % roundoff errors, are related to the Farey sequences. % Key words: roundoff-error, number theory, Farey sequence. % PACS: 6.30C 2.60 (3.30) % Farey sequence, observed in the flow of traffic % (without actually being recognized), H. Reiss, A. D. Hammerich, and E. W. Montroll, J. Stat. Phys. 42, 1986, 647-687. % Farey sequence, observed in the production levels in economics: A. Golan, Mathematical Social Sciences 21, 1991, 261-286. %%%%%%%%%%%%%%%%%%%%% FAREY NUMBERS FINISHED %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% (xy)^2 POTENTIAL %%%%%%%%%%%%%% \bibitem{DR_prl} P. Dahlqvist and G. Russberg, % Existence of stable orbits in the $x^{2}y^{2}$ potential {\em Phys. Rev. Lett. \bf 65}, 2837 (1990). D. Biswas {\em et al.}, %"Existence od stable periodic orbits in $x^2 y^2$ potential: % A semiclassical approach" {\em J. Phys. \bf A 25}, L297 (1992). \bibitem{DR} P. Dahlqvist and G. Russberg, % Periodic orbit quantization of bound chaotic systems, {\em J. Phys. \bf A 24}, 4763 (1991). P. Dahlqvist: ``Semiclassical Mechanics of Bound Chaotic Systems", CHAOS 2, 43 (1992). \bibitem{CarVoz} N. Caranicolas and Ch. Vozikis % numerical work on x^4+y^4+(xy)^2 , {\em Celestial Mechanics . \bf 40}, 35 (1987) C.R. Martens, R.L.Waterland and P. Reinhardt, %Classical, semi-classical and QM versions of (xy)^2 potential spectra {\em J. Chem. Phys. \bf 90 \rm, 2328 (1989). } M. Founargiotakis {\em et al.}, %"Periodic orbits, bifurcations and QM eigenfunctions and spectra" {\em J. Chem. Phys. \bf 91\rm, 1389 (1989). } G. Sohos, T. Bountis and H. Polymilis, %"Is the (xy)^2 potential completely chaotic?" {\em Il Nuovo Cimento, \bf 104 B\rm, 339 (1989). } From: Antti Niemi 9 Mar 2000 There is a lot of discussion $x^{2}y^{2}$ model in string/membrane theory. The classic reference is B. de Wit, M. Luscher and H. Nicolai, Nucl. Phys. B320 (1989) 135 who related it (supersymmetric version) to membranes. % relevant articles can be found by looking at the QSPIRES citations to this. % % a (messy) numerical investigation I. Ya. Aref'eva, P. B. Medvedev, O. A. Rytchkov, I. V.Volovich Chaos in M(atrix) Theory hep-th/9710032 %this article refers to: % \bibitem{BMS} G. Z. Baseyan, S. G. Matinyan and G. K. Savvidi, {\it JETP Lett.} 29(1979)585 \bibitem{Lus} M. L$\ddot{u}$scher, {\it Nucl. Phys.} B219(1983)233 \bibitem{Sim} B. Simon {\it Ann. Phys.} 146(1983)209 \bibitem{Med} B. V. Medvedev, {\it Teor. Mat. Phys.} 60(1984)224; 109(1996)406 \bibitem{WLN} B. de Wit, M.Luscher and H. Nicolai, {\it Nucl. Phys.} B 320 (1989) 135 \bibitem{CS} B. V. Chirikov and D. L. Shepelyanskii, {\it JETP Lett.} 34(1981)164 \bibitem{Shur} E.S. Nikolaevsky and L.N. Shchur, JETP LEtt., {\bf 36} (1982) 218-220; JETP {\bf 58} (1983) 1 \bibitem{Bar} J. D. Barrow and J. Levin, gr-qc/9706065; J. D. Barrow, M. P. Dabrowski, hep-th/9711041 \bibitem{Galt} D. V. Gal'tsov and M. S. Volkov, {\it Phys. Lett.} B256 (1991) 17 %%%%%%%%%%%%%%%%%%%%% (xy)^2 POTENTIAL FINISHED %%%%%%%%%%%%%% %Ericson fluctuations? T. Ericson, PRL 5, 1960 p 430 Mayer-Kuckuk, Ann revs of Nucl Science 16, 183 (1966) , Annals of Phys (NY) 23, 1963 p 390 %correlations: \bibitem{CPR90} F. Christiansen, G. Paladin and H.H. Rugh, {\em Phys. Rev. Lett. \bf 65}, 2087 (1990). \bibitem{CBeck91} C. Beck, %``Higher correlation functions of chaotic dynamical % systems - a graph theoretical approach" {\em Nonlinearity \bf 4}, 1131 (1991). \bibitem{Christiansen90.3} F. Christiansen, S. Isola, G. Paladin and H.H. Rugh, {\em J. Phys. \bf A 23}, L1301 %-L1305 (1990). \bibitem{isola} S. Isola, {\em Comm. Math Phys. \bf 116\rm, 33 (1988).} %Freddy's thesis \bibitem{freddy} F. Christiansen, unpublished. \bibitem{freddy1} F. Christiansen, ``Kaos for Cyklister", {\em Gamma \bf 78\rm, 21 (1989)}. \bibitem{#} Vandewater, W Hoppenbrouwers, M Christiansen, F %``Unstable Periodic-Orbits in the Parametrically Excited Pendulum {\em Phys. Rev. \bf A 1991, Vol 44, Iss 10, pp 6388-6398 \bibitem{selberg} A. Selberg, {\em J. Ind. Math. Soc.} {\bf 20}, 47 (1956). \bibitem{TM-KK} J. Theiler, G. Mayer-Kress and J.B. Kadtke %"Chaotic attractors of locally conservative hyperbolic maps" {\em Physics \bf D}, to appear. \bibitem{myr} P. J. Myrberg, {\em Ann. Acad. Sc. Fenn., Ser. A, }{\bf 259}, 1 (1958). \bibitem{MSS} N. Metropolis, M.L. Stein and P.R. Stein, {\em J. Comb. Theo. } {\bf A15}, 25 (1973). %\bibitem{guc} J. Guckenheimer, {\em Inventions. Math. } {\bf 39}, 165 (1977). %Myrberg trees : %\bibitem{kaw} H. Kawakami and C. Mira, {\em preprint Syst. Dyn. INSA 85-4} %\bibitem{bin_not} For our purposes the $\{0,1\}$ %alphabet is more conveninent than the $\{R,L\}$ %and other customary notations\rf{myr,MSS} %% \rf{myr,MSS,Mira} %because the computer conversion of itineraries to the %corresponding $x$ can in this notation be %implemented by elementary operations on binary strings. \bibitem{moron} P.A.P Moran, {\em Proc. Camb. Phil. Soc. \bf 42},15 (1946). %\bibitem{41.} ??. Jacobson {\em Comm. Math. Phys.} {\bf 81}, 39 (1981). \bibitem{mis1} M. Misiurewicz, {\em Publ. Math. IHES} {\bf 53}, 17 (1981). R. Maeder, {\em Programing in Mathematica} (Addison-Wesley, Redwood CA, 1990). %%%%%%%%%%%%%%%%%%%%%% BERRY-ana %%%%%%%%%%%%%% \bibitem{BM72} M.V. 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Stora, eds, (North Holland, Amsterdam 1983), 171 \bibitem{Ber85} M.V.Berry, {\em Proc. R. Soc (London) Ser.} {\bf A 400}, 229 (1985) \bibitem{Berry86} M.V.Berry, in {\it Quantum Chaos and Statistical Nuclear Physics} (ed. T.H.Seligman and H.Nishioka), {\it Lecture Notes in Physics} {\bf 263}, 1 (Springer, Berlin, 1986). M.V. Berry and R.J. Mondragon, {`` Neutrino Billiards: Time-reversal symmetry-breaking without magnetic fields''} Proc.Roy.Soc.Lond.,A412, 1987, 873-885 \bibitem{KB} J.P. Keating and M.V. Berry, % quantization of a rectangle - periodic orbit sum over % bessels rather than exponentials {\em J. Phys. \bf A 20\rm, L1139 (19??).} \bibitem{BK90} M.V. Berry and J.P. Keating, % A rule for quantizing chaos? - use "our" Selberg products; % conjecture tail resummation -> Riemann Siegel formula {\em J. 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Rev. \bf E 52}, 2463 (1995), DESY report 94-213, chao-dyn 9412007 (1994) \bibitem{BD95} A. Bäcker and H.R. Dullin, {\em ``Symbolic dynamics and periodic orbits for the cardioid billiard''}, DESY report 95-198, chao-dyn 9511004 (1995) R. Aurich, A. Bäcker and F. Steiner, {\em ``Mode fluctuations as fingerprint of chaotic and non-chaotic systems''}, Preprint ULM-TP/96-2. chao-dyn/9608016 %%%%%%%%%%%%%%%%%%%%%% STEINER-ana FINISHED %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% RUELLEIANA %%%%%%%%%%%%%% % Repellers for real analytic maps %"Generalized zeta functions for Axiom A basic sets": \bibitem{ruelle1} D. Ruelle, {\em Bull. Amer. Math. Soc. } {\bf 82}, 153 (1976). \bibitem{grot} A. Grothendieck, {\em ``La th\'eorie de Fredholm", Bull. Soc. Math. France}, {\bf 84}, 319 %- 384 (1956). \bibitem{96.} A. Grothendieck, {\em Produits tensoriels topologiques et espaces nucl\'eaires}, Amer. Meth. Soc. {\bf 16}, Providence R. I. (1955). \bibitem{Plemelj1909} J. Plemelj, {\em ``Zur Theorie der Fredholmschen Funktionalgleichung''}, {\em Monat. Math. Phys. \bf 15}, 93 (1909). % 93-128. \bibitem{Smithies41} F. Smithies, {\em ``The Fredholm theory of integral equations''}, {\em Duke Math. \bf 8}, 107 (1941). % 107-130. \bibitem{TW93} C.A. Tracy and H. Widom, %60 pages LaTeX% % catracy@ucdavis.edu (Craig Tracy) CHECK THIS!: {\em Fredholm Determinants, Differential Equations and Matrix Models}, hep-th/9306042. \bibitem{tak} F. Takens, ``Detecting strange attractors in turbulence", {\em Springer Lecture Notes in Math.} {\bf 898}, 366%-381 (1981). \bibitem{rue68} D. Ruelle, {\em ``Statistical mechanics of a one-dimensional lattice gas''}, {\em Commun. Math. Phys. \bf 9}, 267%--288 (1968). \bibitem{rue_tak} D. Ruelle and F. Takens, \CMP{ 20, 167, 1971} \bibitem{rue_tak2} D. Ruelle and F. Takens, \CMP{ 23, 343, 1971} %\bibitem{6.2} See propositions 6.2, 7.3 of ref. \rf{rueselb}. %repellers for real analy maps \bibitem{grothi} D. Ruelle, {\em Ergod. The. and Dynam. Sys. \bf 2\rm, 99 (1976).} #99-107 \bibitem{ruelle} D. Ruelle, {\em Statistical Mechanics, Thermodynamic Formalism}, (Addison-Wesley, Reading MA, 1978). % Transfer op/Perron Frobenius theory applied % to zeta fns and correlation functions % in the stat mech/Smale space Holder continuous % setting. No Grothendieck theory, no Fredholm dets % because no analyticity. (Essential spectrum % is present.) The hyperbolic case IS % TREATED. \bibitem{ruelle89} D. Ruelle, {\em Chaotic Evolution and Strange Attractors} (Cambridge Univ. Press, Cambridge 1989) \bibitem{EckRue} J.-P. Eckmann and D. Ruelle, {\em Rev. Mod. Phys. \bf 57}, 617 (1985). % definition of the zeta for continuous flow: \bibitem{ruelcont} D. Ruelle, {\em J. Stat. Phys. \bf 44\rm, 281 (1986)}. %on resonances in flows: \bibitem{rue87a} D. Ruelle, {\it J. Diff. Geo.} {\bf 25}, 99 (1987).%99--116 \bibitem{rue87b} D. Ruelle, %``The thermodynamical formalism for expanding maps" {\it J. Diff. Geo.} {\bf 25}, 117 (1987). %117--137 % Two papers were the Gibbs state theory % is constructed for Axiom A flows/diffeos % in the Holder continuous setting. In particular % THIS IS THE THEORETICAL PAPER about the % poles of the Fourier transform of the % correlation function. \bibitem{ruelleele} D. Ruelle, {\em Elements of Differentiable Dynamics and Bifurcation Theory}, (Academic Press, San Diego, 1989). \bibitem{BER} V. Baladi, J.--P. Eckmann, and D. Ruelle, {\em Nonlinearity \bf 2}, 119 (1989). \bibitem{rue89} D. Ruelle, %``The thermodynamical formalism for expanding maps", % IHES preprint P/89/08 (Jan 1989).aRECHECK whether the same paper? {\em Commun. Math. Phys.} {\bf 125}, 239%--262 (1989). %Grothendick error in Ruelle corrected; analogies to Selberg: \bibitem{frie} D. Fried, % ``The Zeta functions of Ruelle and Selberg I" {\em Ann. Scient. \'Ec. Norm. Sup. \bf 19\rm, 491 (1986).} % Grothendieck theory for Axiom A flows % applied to more general zeta and Selberg % functions via Fredholm dets. % No correlation functions % In the two above papers the trick (which I sketched in my last message) % is to quotient out along stable (or unstable) manifolds. The % stable/unstable foliation must hence be assumed real analytic % in order to get a real analytic expanding system from the % hyperbolic real analytic one. This hypothesis is very strong - \bibitem{Ruelle90} D. Ruelle, {\em ``An extension of the theory of Fredholm determinants"}, {\em Inst. Hautes \'Etudes Sci. Publ. Math. \bf 72}, 175-193 (1990). \bibitem{Ruelle95} D. Ruelle, {\em ``Functional determinants related to dynamical systems and the thermodynamic formalism}, preprint IHES/P/95/30 (March 1995). \bibitem{Rue94} D. Ruelle, {\em Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval}, %{\rm CRM Monograph Series, Vol. 4} (Amer. Math. Soc., Providence, NJ 1994) \bibitem{Haydn90} N. Haydn, {\em Comm. Math. Phys. \bf 134}, 217-236 (1990). % An analogue of the prime number theorem for closed % rbits of Axiom A flows: \bibitem{PP} W. Parry and M. Pollicott, {\em Ann. Math. \bf 118\rm, 573 (1983).} \bibitem{Parry86} W. Parry, {\em Comm. Math. Phys. \bf 106}, 267 (1986). \bibitem{Pollicott85} M. Pollicott, {\em Invent. Math. \bf 81}, 413-26 (1985). % {\em On the rate of mixing of Axiom A flows} \bibitem{polli} M. Pollicott, {\em ``Meromorphic extensions of generalised zeta functions''}, {\em Invent. Math. \bf 85}, 147 (1986). %147--164 \bibitem{PP90} W. Parry and M. Pollicott, {\em Zeta Functions and the periodic Structure of Hyperbolic Dynamics}, {\em Ast\'erisque {\bf 187--188}} (Soci\'et\'e Math\'ematique de France, Paris 1990). \bibitem{Pollicott90} M. Pollicott, {\em Ann. of Math. \bf 131}, 331-354 (1990). % {\em shows gap in spectrum of hyperbolic systems} \bibitem{Pollicott91} M. Pollicott, ``A Note on the Artuso-Aurell-Cvitanovi\'c approach to the Feigenbaum tangnt operator'', {\em J. Stat. Phys., \bf 62\rm, 257 (1991)}. \bibitem{Jiang} Y. Jiang, T. Morita and D. Sullivan, ``Expanding direction of the period doubling operator'', {\em Comm. Math. Phys. \bf 144}, 509 (1992). % 509--520 % SUNY Stony Brook Inst. for Math. Sci. prperint No 1991/6. \bibitem{dorfle} D\"orfle M 1985 {\em J. Stat. Phys.} {\bf 40} 93%--132 \bibitem{Weil49} A. Weil, {\em ``Numbers of solutions of equations in finite fields''}, {\em Bull. Am. Math. Soc. \bf 55}, 497 (1949). \bibitem{AM} The $\zeta$ functions were originally introduced in this context by E. Artin and B. Mazur, {\em Annals. Math. } {\bf 81}, 82 (1965). See for example ref.~\cite{MT} for their evaluation for maps of the interval. \bibitem{MT} J. Milnor and W. Thurston, "On iterated maps of the interval", in A. Dold and B. Eckmann, eds., {\em Dynamical Systems, Proceedings, U. of Maryland 1986-87, Lec. Notes in Math. \bf 1342, \rm 465 (Springer, Berlin 1988)}. % rational zeta for finite subshift type: ref in Nordahl's thesis \bibitem{bowlan} R. Bowen and O.E. Lanford {\em Math. ??} {\bf ??}, \bibitem{bowlan1} R. Bowen and O.E. Lanford, {\em ``Zeta functions of restrictions"}, pp. 43-49 in {\em Proceeding of the Global Analysis}, (A.M.S., Providence 1968). \bibitem{manning} A. Manning, ``Axiom A diffeomorphisms have rational zeta function'', {\em Bull. London Math. Soc.\bf 3}, 215 (1971). % 215-220. %accurate interval sums for escape rates; evidence of %zeta fct pole collisions \bibitem{Kov_Tel} Z. Kov{\'a}cs and T. T{\'e}l, preprint Aug. 1989. \bibitem{SMYO} Shigematsu H, Mori H, Yoshida T and Okamoto H 1983 {\em J. Stat. Phys.} {\bf 30} 649%-- \bibitem{Hurt} N.E. Hurt, ``Zeta Functions and Periodic Orbit Theory: A Review", {\em Results in Mathematics \bf 23}, 55 %-120, (Birkh\"a user, Basel 1993). \bibitem{Kitchens98} B.P. Kitchens, Symbolic dynamics: one-sided, two-sided, and countable state Markov shifts (Springer, Berlin 1998). \bibitem{Kitchens} B.P. Kitchens, ``Symbolic dynamics, group automorphisms and Markov partition", in {\em Real and Complex Dynamical Systems}, B. Branner and P. Hjorth, ed. (Kluwer, Dordrecht, 1995). \bibitem{Baladi} V. Baladi, ``Dynamical Zeta Functions", in {\em Real and Complex Dynamical Systems}, B. Branner and P. Hjorth, ed. (Kluwer, Dordrecht, 1995). \bibitem{hejhal} D.A. Hejhal, {\em Duke Math. J. \bf 43\rm, 441 (1976).} \bibitem{EA89} B. Eckhardt and E. Aurell, {\em Europhys. Lett. {\bf 9}, 509 (1989)} % or (1991)?? % abcissa of abyssimal convergence R.F. Williams % ``A new zeta function, natural for links'' Proceedings of the Smalefest (Berkeley, CA 1990), Springer, NY, 1993. S. Kennedy, M. Stafford, and R. Williams, %{\em A new Cayley-Hamilton Theorem}, to appear. % B. Eckhardt\rf{EG} credits refs.~\cite{TS86,FI87} % with introducing the generalized evolution operators. % PC: I have looked at them: \bibitem{TS86} T. T\'el and P. Szepfalusy, {\em Phys. Rev. \bf A 34}, 387 (1986). % only has a brief comment that $\langle e^{\alpha(x)} \rangle$ % could be used to compute escape rates \bibitem{FI87} H. Fujisaka and M. Inoue, {\em Progr. Theor. Phys.} {\bf 78}, 268 (1987). % indeed explicitely write down the Ruelle operator, but % strangely enough never refer to Ruelle. My attitude is % (at least for a brief paper) it is sufficient to % refer to Ruelle, who always defines his zeta functions % weighted by a multipicative $g(x)$ factor; later papers % only rediscover that, but do not use zeta functions and cycle % expansions, so do not contribute significantly new stuff - % seems to me.... %%%%%%%%%%%%%%%%%%%%%% RUELLEIANA FINISHED %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% GASPARD %%%%%%%%%%%%%% % deals with intermitency type power laws: \bibitem{GW88} P. Gaspard and X.-J. Wang % ``Sporadicity: between periodic and chaotic dynamical behaviors" {\em Proc. Natl. Acad. Sci. USA, \bf 85}, 4591 (1988). \bibitem{gasp} P. Gaspard and S.A. Rice, %\bibitem{gr_cl} P. Gaspard and S.A. Rice, {\em Scattering from a classically chaotic repellor}, {\em J. Chem. Phys. \bf 90}, 2225 (1989); % 2225--2241. %\bibitem{gr_sc} P. Gaspard and S.A. Rice, {\em Semiclasscial quantization of the scattering from a classically chaotic repellor}, %J. Chem. Phys. {\bf 90} (1989) 2242--2254. {\bf 90}, 2242 (1989); {\em Exact quantization of the scattering from a classically chaotic repellor}, %{\em J. Chem. Phys. \bf 90}, 2256 (1989); %-2262. {\bf 90}, 2255 (1989). \bibitem{GA} P. Gaspard and D. Alonso Ramirez, % "Ruelle classical resonances and dynamical chaos: % The three- and four-disk scatterers" {\em Phys. Rev. \bf A 45}, 8383 (1992). \bibitem{alonso} P. Gaspard and D. Alonso, {\em ``$\hbar$ expansion for the periodic-orbit quantization of hyperbolic systems''}, {\em Phys. Rev. \bf A 47}, R3468 (1993); % R3468-3471 D. Alonso and P. Gaspard, {\em `` $\hbar$ expansion for the periodic orbit quantization of chaotic systems''}, {\em CHAOS \bf 3}, 601 (1993). % 601-612. \bibitem{gasp_hbar} P. Gaspard, {\em ``$\hbar$-Expansion for quantum trace formulas''}, in G. Casati and B. Chirikov, eds., {\em Quantum chaos between order and disorder} (Cambridge University Press 1995), pp.~385-404. % Entropy+Lyapunovs --> transport coefficients: \bibitem{ga2}P. Gaspard, Phys.Lett. A {\bf 168}, 13 (1992); Chaos {\bf 3}, 427 (1993) %diffusion coefficient for Lorentz gas: \bibitem{GN} P. Gaspard and G. Nicolis, %"Transport properties, Lyapunov exponents, and entropy per % unit time" {\em Phys. Rev. Lett. \bf 65}, 1693 (1990). \bibitem{gasp91} P. Gaspard %Diffusion, effusion and chaotic scattering: %an exactly solvable Liouvillian dynamics % - his baker map calculation Univ. Libre de Bruxelles preprint (Feb. 1991). {\em J. Stat. Phys.} {\bf 68}, 673 (1992). \bibitem{GB92} P. Gaspard and F. Baras, in M. Mareschal and B.L. Holian, eds., {\em Microscopic simulations of Complex Hydrodynamic Phenomena} (Plenum, NY 1992). \bibitem{jrd}, P. Gaspard and J.R. Dorfman, ``Chaotic scattering theory, ...'', {\em Phys. Rev. \bf E 52}, 3525 (1995). \bibitem{GNPT} P. Gaspard, G. Nicolis, A. Provata and S. Tasaki ``Spectral signature of the pitchfork bifurcation: Liouville equation approach'' {\em Phys. Rev. \bf E 51}, 74 (1995). \bibitem{#} Gaspard, P %``R-Adic One-Dimensional Maps and the Euler Summation Formula {\em J. Phys. \bf A 1992, Vol 25, Iss 8, pp L483-L485 \bibitem{PG97} P. Gaspard, {\em Chaos, Scattering and Statistical Mechanics} (Cambridge Univ. Press, Cambridge 1997). %%%%%%%%%%%%%%%%%%%%%% GASPARD FINISHED %%%%%%%%%%%%%% % VATTAY: a nice paper about the quantum potential and dynamics in it. {\em Phys. Lett. A \bf 183} 413 (1993) %%%%%%%%%%%%%%%%%%%%%% FREDHOLMS %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% FREDHOLMS FINISHED %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% HHRUGH THESIS Jul 14 1992 %%%%%%%%%%%%%% \bibitem{Cartan61} H. Cartan, {\em Th\'eorie \'el\'ementaire des fonctions analytiques d'une ou plusieurs variables complexes}, Hermann, Paris (1961). \bibitem{Dunford57} N. Dunford and J. Schwartz,{\em Linear Operators,} Interscience Publishers, INC., New York (1957). \bibitem{Herve63} M. Herv\'e, {\em Several complex variables, local theory}, Oxford University Press (1963). \bibitem{Reed72} M. Reed and B. Simon, {\em Methods of Modern Mathematical Physics,} Academic Press, INC. (1972). \bibitem{Riesz55} F. Riesz and B. Sz.-Nagy, {\em Functional analysis}, Frederick Ungar Publ. Co. (1955). \bibitem{Rugh92} H.H. Rugh, {\em ``The Correlation Spectrum for Hyperbolic Analytic Maps"}, {\em Nonlinearity \bf 5}, 1237 (1992). \bibitem{Rugh93} H.H. Rugh, {\em Correlation spectra for Anosov diffeomorphisms and billiards}, {in preparation}. %%%%%%%%%%%%%%%%%%%%%% HHRugh FINISHED %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% ACOUSTICS %%%%%%%%%%%%%%%%%%%%%% Visscher, Migliori, Bell, Reinert %``On the normal modesof free vibration of inhomogenous and anisotropic elastic objects'' {\em J. Acoust. Soc. Am. \bf 90}, 2154 (1991) %M. Berry recommends: Weyl formula for acoustics Balters and Hilf, {\em Spectra of Finite Systems} (Birkha\"user 1976) %M. Berry recommends: Weyl formula for acoustics Balian and Bloch (1971) %M. Berry recommends: ``optically'' active media? Musgrave, Proc. Roy Soc. (sometime in 1980's) %M. Berry recommends: Chruch Bells: Bob Perrin, Physics, Loughborough University, U.K. C.L. Pekeris, ``Seismic buried pulse'', {\em Proc. Natl. Acad. Sci. USA, \bf 41}, 469-80 (1955). J.B. Keller and ?. Carrol, ``Geometrical theory of elastodynamic...'' {\em J. Acoust. Soc. Am. \bf 36}, 32-40 (1964) % Keller: do not need to read thisone, it is included in the above: F. Gilbert ``Scattering of impulse elastic...'' {\em J. Acoust. Soc. Am. \bf 32}, 841-857 (1960) C.L. Pekeris and H. Lifson, ``Motion of the surface of a uniform elastic half-space produced by a buried pulse'', {\em J. Acoust. Soc. Am. \bf 29}, 1233-20 (1957) % Niels Sondergaard 7 Oct 2002: % Hvis man bruger Keller's teorier naivt saa naar man ikke saerlig langt: \bibitem{Rulf69} B. Rulf, ``Rayleigh Waves on Curved Surfaces", {\em J. Acoust. Soc. \bf 45}, 493 (1969). \bibitem{Kudrolli02} T. Neicu and A. Kudrolli, % ``Periodic orbit analysis of an elastodynamic resonator % using shape deformation,'' {\em Europhys. Lett. \bf 57}, 341 (2002); % 341-347 {\tt cond-mat/0110543}. %%%%%%%%%%%%%%%%%%%%%% ACOUSTICS FINISHED %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% SEMICLASSICS %%%%%%%%%%%%%%%%%%% \bibitem{Ikawa} M. Ikawa, {\em On poles of scattering matrices for several convex bodies}, {\em Osaka J. Math., \bf 27\rm, 281 (1990)} \bibitem{Shudo} A. Shudo, %``Length Spectrum and Semiclassical Density of States for % an Almost-Integrable Billiard System {\em Phys. Rev. \bf A 46}, 809 (1992). % pp 809-824 \bibitem{HS92} T. Harayama and A. Shudo, % ``Periodic orbits and semiclassical quantization of dispersing % billiards" {\em J. Phys. \bf A 25}, 4595 (1992). \bibitem{KHD} M. Ku\v{s}, F. Haake and D. Delande, % ``Prebifurcation periodic ghost orbits in semi-classical quantization'', {\em Phys. Rev. Lett. \bf 71}, 2167 (1993). \bibitem{aw_chaos} A. Wirzba, % ``Validity of the semiclassical periodic orbit % approximation in the 2-and 3-disk problems", {\em CHAOS \bf 2}, 77 (1992). A. Wirzba and P.E. Rosenqvist, ``Three Disks in a Row: A Two-Dimensional Scattering Analog of the Double-Well Problem'', % \> TH Darmstadt preprint (1996), IKDA 96/8, {\em Phys. Rev.\bf A 54}, 2745 (1996). % 2745--2754, Erratum: {\em Phys. Rev. \bf A 55}, 1555 (1997); {\tt chao-dyn/9603016.} \bibitem{Watson} G.N. Watson, {\em Proc. Roy. Soc. London Ser. \bf A 95}, 83 (1918). \bibitem{aw_nucl} A. Wirzba, {\it Test of the Periodic Orbit Approximation in n-Disk Systems,} {Nucl. Phys.} {\bf A560} (1993) 136-150. \bibitem{VWR94} G. Vattay, A. Wirzba and P.E. Rosenqvist, %{\em Periodic Orbit Theory of Diffraction}, ``Periodic Orbit Theory of Diffraction'', {\em Phys. Rev. Lett. \bf 73}, 2304 (1994). \bibitem{vwr_japan} G. Vattay, A. Wirzba and P.E. Rosenqvist, {\em Inclusion of Diffraction Effects in the Gutzwiller Trace Formula}, Proceedings of the International conference on {\sc Dynamical Systems and Chaos\/}, eds.\ Y.~Aizawa, S.~Saito and K.~Shiraiwa, (World Scientific, Singapore 1995),, Vol. 2, pp.~463, {\tt chao-dyn/9408005}. \bibitem{vwr_stat} P.E. Rosenqvist, G. Vattay and A. Wirzba, {\em Application of the Diffraction Trace Formula to the Three Disk Scattering System}, {\em J. Stat. Phys. \bf 83}, 243 (1996) %243-257. \bibitem{BAGOP} R. Bl\"u mel, T.M. Antonsen, Jr., B. Georgeot, E. Ott and R.E. Prange, ``Ray splitting and quantum chaos'', U. of Maryland preprint 1995. %creeping orbits in soft potentials were studied by: Kirk W. Mcvoy University of Wisconsin at Madison address: 1150 UNIV AVE MADISON, WI 53706 building: CHAMBERLIN HALL, THOMAS C 5279 phone: 608-262-1152 phone2: 608-262-8186 email: MCVOY%WISCPSL.BITNET@MACC.WISC.EDU title: PROFESSOR EMER department: PHYSICS \bibitem{#} Strunz, WT %``Semiclassical Path Representation of the Green-Function in One-Dimensional Multiple-Well Potentials {\em J. Phys. \bf A 1992, Vol 25, Iss 13, pp 3855-3872 E.J. Heller \jour Phys.Rev.Lett. \vol 53 \pages 1515 (1984) \bibitem{#} Oconnor, PW Tomsovic, S Heller, EJ %``Accuracy of Semiclassical Dynamics in the Presence of Chaos {\em J. Stat. Phys. \bf 1992, Vol 68, Iss 1-2, pp 131-152 \bibitem{#} Sepulveda, MA Tomsovic, S Heller, EJ %``Semiclassical Propagation - How Long Can It Last {\em Phys. Rev. Lett. \bf 1992, Vol 69, Iss 3, pp 402-405 \bibitem{#} Sridhar, S Heller, EJ %``Physical and Numerical Experiments on the Wave Mechanics of Classically Chaotic Systems {\em Phys. Rev. \bf A 1992, Vol 46, Iss 4, pp 1728-1731 \bibitem{#} Heller, EJ %``Semiclassical Time Evolution Without Root Searches - Comments and Perspective - Reply {\em JOURNAL OF CHEMICAL PHYSICS 1991, Vol 95, Iss 12, pp 9431-9432 \bibitem{Schroed26} E. Schr\"odinger, {\em Annalen der Physik \bf 79}, 361, 489; {\bf 80}, 437, {\bf 81}, 109 (1926). \bibitem{Mad1926} E. Madelung, {\em Z. f. Physik \bf 40}, 332 (1926). \bibitem{Bohm52} D. Bohm, %``A suggested Interpretation of the Quantum Theory in Terms % of ``Hidden'' Variables. I'' {\em Phys. Rev. \bf 85}, 166 (1952). \bibitem{PV95} R.H. Parmenter, R.W. Valentine ``Deterministic chaos and the causal interpretation of quantum mechanics'' {\em Phys. Lett. \bf A 201}, 1 (1995). \bibitem{holland93} P.R. Holland, {\em The quantum theory of motion} {\em - An account of the de Broglie-Bohm casual interpretation of quantum mechanics} (Cambridge Univ. Press, Cambridge 1993). Jammer, Max; The philosophy of quantum mechanics. The interpretations of quantum mechanics in historical perspect, Wiley, New York, 1974, p.536 Jammer, Max; The conceptual developement of quantum mechanics. McGraw-Hill, New York, 1966, p.399 \bibitem{dyson52} F. J. Dyson, ``Divergence of Perturbation Theory in Quantum Electrodynamics'', {\em Phys. Rev. \bf 85}, 631 (1952). % pp. 631--632. \bibitem{BGS84b} O. Bohigas, M.J. Giannoni and C. Schmit, {\em J. Phys. Lett. \bf 45}, L1015 (1984). %Effective potential techniques: \bibitem {1.} P. Stevenson, a){\sl Phys. Rev. D} {\bf 30}, 1712 (1984); b){\sl ibid} {\bf 32}, 1389 (1985) and references therein. %Effective potential techniques: \bibitem {2.}a) A.K. Pattanayak and W.C. Schieve, {\sl Phys. Rev. A} {\bf 46}, 1821 (1992); b) L. Carlson and W.C. Schieve, {\sl Phys. Rev. A} {\bf 40}, 5896 (1989); c) A.K. Pattanayak and W. C. Schieve, {\em Semiquantal dynamics of fluctuations: ostensible quantum chaos } (Paper: chao-dyn/9404001, 1994) %From: ARJENDU@UTAPHY.PH.UTEXAS.EDU %Date: Fri, 15 Apr 1994 %a review of EBK quantization: \bibitem {3.} I.C. Percival, {\sl Adv. Chem. Phys. }{\bf 36}, 1 (1977) %Mean-field theories for QM systems: \bibitem {9.}F. Cooper, S.-Y. Pi and P.N. Stancioff, {\sl Phys. Rev. D} {\bf 34}, 3831 (1986). \bibitem {10.}A. Kovner and B. Rosenstein, {\sl Phys. Rev.} {\bf D39}, 2332, (1989). %Mean-field theories for QM systems - reviews: \bibitem {12.}W.-M. Zhang, D.H. Feng and R. Gilmore, {\sl Rev. Mod. Phys. }{\bf 62}, 867 (1990). \bibitem {20.}Y. Tsue, {\sl Prog. Theor. Phys. }{\bf 88}, 911 (1992) and references therein. %time-dep. variational principle \bibitem {8.}R. Jackiw and A. Kerman, {\sl Phys. Lett. }{\bf 71A}, 158 (1979). \bibitem {13.}P.A.M. Dirac, Appendix to the Russian edition of 'The Principles of Quantum Mechanics', as cited by Frenkel, I.I., 'Wave Mechanics, Advanced General Theory' (Clarendon Press, Oxford, 1934) (pg. 253, 436). \bibitem {14.}J. Klauder,{\sl J. Math. Phys.},{\bf 4}, 1055 (1963); 1058 (1963). %coherent states: \bibitem {11.}J. Klauder and B.-S. Skagerstam, 'Coherent States: Applications in Physics and Mathematical Physics' (World Scientific, 1985). \bibitem {15.}E.C.G. Sudarshan, {\sl Phys. Rev Lett.}{\bf 10},277 (1963) \bibitem {16.}R. Glauber, {\sl Phys. Rev. Lett. }{\bf 10},84 (1963); {\sl Phys. Rev. }{\bf 130}, 2529 (1963). \bibitem {17.}A.M. Perelmov, 'Generalized Coherent States and their Applications' (Springer-Verlag, 1986) % classical limit QM coherent states: Berry phase=Bohr-Sommerfeld: \bibitem {18.}R.G. Littlejohn, {\sl Phys. Rev. Lett. }{\bf 61}, 2159 (1988). % Berry phase: \bibitem {19.}M.V. Berry,{\em Proc. Roy. Soc.} {\bf A392}, 45 (1984); A. Aharanov and J. Anandan, {\sl Phys. Rev. Lett.} {\bf 56}, 2000 (1986). %truncated coherent states --> semiclassics: \bibitem {21.}E.J. Heller, {\em J. Chem. Phys.} {\bf 62}, 1544 (1975); in 'Chaos and Quantum Physics', Proceedings of the Les Houches Summer School 1989, (North-Holland, 1991). \bibitem {22.} W.-M. Zhang and D.H. Feng , 'Quantum Nonintegrability', {\sl Phys. Rep.} (In press, 1994); {\sl Mod. Phys. Lett}{\bf A8},1417 (1993). \bibitem {25.}A. Wolf, J.B. Swift, H.L. Swinney and J.A. Vastano, {\sl Physica } {\bf 16D}, 285 (1985). \bibitem {26.}G.M. Zaslavsky, R.Z. Sagdeev,D.A. Usikov and A.A. Chernikov, 'Weak Chaos and Quasi-regular Patterns' (Cambridge University Press, 1991). % is ther Q chaos? \bibitem {27.}A. Heslot, {\sl Phys. Rev. D } {\bf 31}, 1341 (1985). % be careful wtih Quantum fluctuations: \bibitem {28.}R. Balian and M. V\'en\'eroni,{\sl Ann. Phys.} (N.Y.) {\bf 87}, 29 (1988). \bibitem{BW96} H. Bruus and N.D. Whelan, {\em``Edge diffraction, trace formulae and the cardioid billiard''}, {\em Nonlinearity \bf 9}, 1023 (1996), chao-dyn/9509005. %%%%%%%%%%%%%% SEMICLASSICS FINISHED %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% GUTZWILLERiana %%%%%%%%%%%%%%%%%%%%%% \bibitem{gut67} M.C. Gutzwiller, {\em J. Math. Phys. \bf8\rm, 1979 (1967), \bf10\rm, 1004 (1969), \bf11\rm, 1791 (1970).} \bibitem{gut71} M.C. Gutzwiller, {\em J. Math. Phys. \bf12\rm, 343 (1971).} %AniKepler as chaotic system first introduce? recheck! %spectrum for only the deformed circular orbit \bibitem{gut73} M.C. Gutzwiller, {\em J. Math. Phys. \bf14\rm, 139 (1973).} %folliation first drawn? %supposedly contains Gutzwiller scattering on hyperb. surfaces: \bibitem{FP} L.D. Fadeev and B.S. Pavlov, %"Scattering Theory and Automorphic Functions" {\em Seminar of Steklov Mathematical Inst. Lenigrad \bf 27}, 161 (1972). \bibitem{BB72} R. Balian and C. Bloch, \AP{63}, 592 (1971) \bibitem{bb_2} R. Balian and C. Bloch, {\em Distribution of eigenfrequencies for the wave equation in a finite domain: III. Eigenfrequency density oscillations}, {\em Ann. Phys. (N.Y.) \bf 69},76 (1972). % 76-160. \bibitem{bb_1} R. Balian and C. Bloch, {\em Solution of the Schr\"{o}dinger Equation in Terms of Classical Paths} {\em Ann. Phys. (NY) \bf 85}, 514 (1974). % 514-545. \bibitem{gut77} M.C. Gutzwiller, {\em J. Math. Phys. \bf18\rm, 806 (1977).} %AniKepler symbolic dynamics %collision manifolds \bibitem{dev78} R.L. Devaney, %AniKepler symbolic dynamics {\em J. Diff. Equ.} {\bf 29}, 253 (1978); {\em Inventiones math. \bf 45\rm, 221 (1978).} % this seems wrong reference?: % {\em Springer Lecture Notes in Math.} {\bf 597}, 271 (1977). \bibitem{dev78a} R.L. Devaney, %Transverse heteroclinic orbits in the anisotropic Kepler problem in {\em The structure of Attractors in Dynamical Systems}, {\em Springer Lecture Notes in Math.} {\bf 668}, (1978). \bibitem{devaney_nitecki}R.L. Devaney and Nitecki, {\em Com. Math. Phys.} {\bf 67}, 137 (1979). \bibitem{G79} M.C. Gutzwiller, in {\it Stochastic Behavior in Classical and Quantum Hamiltonian Systems}, ed. G.Casati and J.Ford (Springer, Berlin 1979), p.316. % on Hadamard billiards?: \bibitem{gut80} M.~C.~Gutzwiller, \PRL{45}, 150 (1980);% hyperbolic space; first AK spectrum publ. \PST{9}, 184 (1985); \CM{53}, 215 (1986). \bibitem{gut81} M.C. Gutzwiller, in R.L. Devaney and Z.H. Nitecki, {\em Classical Mechanics and Dynamical Systems} (Marcel Dekker, New York 1981), pp. 69-90. % list of orbits of length 10 \bibitem{gut82} M.C. Gutzwiller, {\em Physica \bf D5\rm, 183 (1982).} %"The quantization of a classically ergodic system" %uses Ising model to improve summations %introduces symetrizations of the spectra %computes the spectrum \bibitem{gut84} M.C. Gutzwiller, {\em J. Phys. Chem. \bf92\rm, 3154 (1984).} % his most detailed reference on zeta functions \bibitem{gutbook} M.C. Gutzwiller, {\em Chaos in Classical and Quantum Mechanics} (Springer, New York 1990). %this review has all our factorizations for the octogonal %hyperbilliard case, eq (VII.47). \bibitem{BV} N. Balasz and A. Voros, {\em Chaos on the Pseudosphere, Phys. Rep. \bf 143}, 109 %--240 (1986). % this should talk about "pseudo zeros"? \bibitem{voros88} A. Voros, {\em Unstable periodic orbits and semiclassical quantisation}, {\em J. Phys. \bf A 21\rm, 685 (1988).} \bibitem{7} A. Voros, in: {\it The Riemann Problem, Complete Integrability and Arithmetic Applications\/}, eds. D. Chudnovsky and G. Chudnovsky, Lecture Notes in Mathematics {\bf 925}, Springer, Berlin (1982) p.184-208 (augmented version of: Nucl. Phys. {\bf B165} (1980) 209-236). [Hint: $Z^\pm (s)$ (here) $\equiv {C_M^{-{2M \over M+1}s} (Z(s) \pm Z^{\rm P}(s))/2}$ (in~[7]).] % A readable introduction is given in \bibitem{voros87} A. Voros, {\em Spectral Functions, Special Functions and the Selberg Zeta Function}, {\em Commun. Math. Phys. \bf 110}, 439 (1987). % 439-465. \bibitem{9} A. Voros, {J. Physique-LETTRES \bf 43} (1982) L1-L4; A. Voros, in: {\it Zeta Functions in Geometry\/} (Proceedings, Tokyo 1990), eds. N. Kurokawa and T. Sunada, Advanced Studies in Pure Mathematics {\bf 21}, Math. Soc. Japan, Kinokuniya, Tokyo (1992), p.327-358. \bibitem{V94} Andr\'e Voros, %From voros@amoco.saclay.cea.fr Thu Mar 10 11:25 MET 1994 ``EXACT QUANTIZATION CONDITION FOR ANHARMONIC OSCILLATORS (in one dimension)'', Saclay preprint T94/028 (March 1994) submitted to J. Phys. A (Letters) \bibitem{G87} M.C.Gutzwiller, J. Phys. Chem. {\bf 92}, 3154 (1987). \bibitem{gut88} M. Gutzwiller, % unstable manifold foliation paper {\em J. Chem. Phys. ?? (1988).} \bibitem{gut89} M.C. Gutzwiller, {\em Physica \bf D38\rm, 160 (1989).} %Multifractal measures and stability islands in the aniKep %numerical evidence that there are no Broucke islands for mu>2 \bibitem{RGL95} R.G. Littlejohn, ``Semiclassical structure of trace formulas'', in G. Casati and B. Chirikov, eds., {\em Quantum Chaos}, (Cambridge University Press, Cambridge 1994). \bibitem{DHN74} R. Dashen, B. Hasslacher and A. Neveu , ``Nonperturbative methods and extended hadron models in field theory. 1. Semiclassical functional methods.'', {\em Phys. Rev. \bf D10}, 4114 (1974). %%%%%%%%%%%%%%%%%%%%%% GUTZWILLERIANa FINISHED %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% QM ZETAS %%%%%%%%%%%%%%%%%%%%%% \bibitem{57.} {\norm B. Eckhardt}, {\em Quantum Mechanics of Classically Non-Integrable Systems}, Physics Reports, {\bf 163:4} {\norm 207-297 }, {\norm (1988)}. \bibitem{58.} {\norm P. Pechukas}, {\em Remarks on ``Quantum Chaos"}, J. Chem. Phys., {\bf 88} 4823-4829, (1984). \bibitem{laur} B. Lauritzen, % Discrete Symmetries and the Periodic Orbit Expansions {\em Phys. Rev. \bf A 43}, 603 (1991). %It is shown that the periodic orbits that are point-wise %invariant under a symmetry operation require a special treatment. \bibitem{DettDahl} C.P. Dettmann and P. Dahlqvist, {\em Computing the diffusion coefficients for intermittent maps - Resummation of stability ordered cycle expansions }, {\em Phys. Rev. \bf E 57}, 5303 (1998). \bibitem{Delos1} J.B. Delos, {\em Adv. Chem. Phys.} {\bf 65}, 161 (1986) \bibitem{Delos2} J.B. Delos, \JCP{86}, 425 (1987) \bibitem{Stoe} J. Stein and H.J. St\"ockmann, \PRL{64}, 2215 (1990) \bibitem{IS_rev2} U. Smilansky in Les Houches Summer School \bibitem{Rob} J.M. Robbins, \PRA{40}, 2128 (1989) \bibitem{Stoe} J. Stein and H.J. St\"ockmann, \PRL{64}, 2215 (1990) \bibitem{Bogo91a} E.B. Bogomolny, {\em Comm. At. Mol. Phys.} {\bf 25}, 67 (1990). \bibitem{Bog92} E.B. Bogomolny, CHAOS {\bf 2}, 5 (1992). \bibitem{Bogo92} E.B. Bogomolny, {\em Nonlinearity \bf 5}, 805 (1992). % 805--866 (1992). \bibitem{BogCar93} E.B. Bogomolny and M. Caroli, {\em Physica \bf D 67}, 88 (1993). \bibitem{DoSmi91} E. Doron and U. Smilansky, in \cite{CHAOS92} \bibitem{#} Doron, E Smilansky, U %``Chaotic Spectroscopy {\em Phys. Rev. Lett. \bf 1992, Vol 68, Iss 9, pp 1255-1258 \bibitem{#} Doron, E Smilansky, U Frenkel, A %``Chaotic Scattering and Transmission Fluctuations {\em Physica \bf D 1991, Vol 50, Iss 3, pp 367-390 \bibitem{BH} H.P. Baltes and E.R. Hilf, {\em Spectra of finite systems} BI Wissenschaftsverlag, Mannheim 1976 \bibitem{Ikawa} M. Ikawa, {\em On poles of scattering matrices for several convex bodies}, preprint Osaka 1990 \bibitem{KL} W. Kohn and J.M. L\"uttinger, {\em Phys. Rev. } {\bf 96}, 1488 (1954). %test of GOE for AniKepler (neither GOE nor Poisson) \bibitem{KKR} W. Kohn and N. Rostoker, {\em Solution of the Schr\"{o}dinger equation in periodic lattices with an application to methalic lithium}, Phys. Rev. {\bf 94} (1954) 1111-1120; J. Korringa, Physica {\bf 13} (1947) 392-400. \bibitem{faulk} R.A. Faulkner, {\em Phys. Rev. } {\bf 184}, 713 (1969). %QM computations of the AniKepler-labeling convention introd. \bibitem{Wi87} D. Wintgen, \PRL{58}, 1589 (1987) \bibitem{wint88} D. Wintgen, H. Marxer and J.S. Briggs, {\em J. Phys. \bf A} {\bf 20}, L965 (1987). %QM computations of the AniKepler spectrum - 1000 lines \bibitem{wint88a} D. Wintgen and H. Marxer, {\em Phys. Rev. Lett.} {\bf 60}, 971 (1988). %test of GOE for AniKepler (neither GOE nor Poisson) \bibitem{wint88b} D. Wintgen {\em Phys. Rev. Lett.} {\bf 61}, 1803 (1988). \bibitem{WF87} D. Wintgen and H. Friedrich {\em Phys. Rev.} A{\bf 36}, 131 (1987). \bibitem{ERTW} G.S. Ezra, K. Richter, G. Tanner and D. Wintgen, % Semiclassical Cycle Expansion for the Helium Atom {\em J. Phys. \bf B 24}, L413 (1991). \bibitem{GSBEC} G. Tanner, P. Scherer, E.B. Bogomolny, B. Eckhardt and D. Wintgen, %``Quantum Eigenvalues from Classical Periodic-Orbits {\em Phys.Rev.Lett.} {\bf 67} (1991) 2410 \bibitem{wint92} D. Wintgen, K. Richter and G. Tanner, % `` The semiclassical helium atom", {\em CHAOS \bf 2}, 19 (1992). \bibitem{RTW91} K. Richter, G. Tanner and D. Wintgen, in \cite{CHAOS92} \bibitem{HDF93} G. Handke, M. Draeger and H. Friedrich, ``Classical dynmamics of s-wave helium'', {\em Physica \bf A 197}, 113 (1993). \bibitem{HDIF93} G. Handke, M. Draeger, W. Ihra and H. Friedrich, ``Scattering and($e$,2$e$) reactions in classical s-wave helium'', {\em Phys. Rev.} A{\bf 48}, 3699 (1993). \bibitem{Handke94} G. Handke, ``Fractal dimensions in the phase space of two-electron atoms'', {\em Phys. Rev.} A{\bf 50}, R3561 (1994). \bibitem{HDIF94} M. Draeger, G. Handke, W. Ihra and H. Friedrich, ``one- and two-electron excitations of helium the s-wave model'', {\em Phys. Rev.} A{\bf 50}, 3793 (1994). \bibitem{TW} G. Tanner and D. Wintgen, % Quantization of chaotic systems {\em CHAOS \bf 2}, 53 (1992). \bibitem{TW91} G. Tanner and D. Wintgen, in \cite{CHAOS92} \bibitem{massAKep} B.W. Levinger and D.R. Frankel, {\em J. Phys. Chem. Solids \bf 20}, 281 (1961); J.C. Hensel, H. Hasegawa and M. Nakayama, {\em Phys. Rev. \bf 138}, 225A (1965). %experimental measurement of mass anisotrpy for AniKepler \bibitem{NHK} H. Navarro, E.E. Haller and F. Keilmann, {\em Phys. Rev. \bf B } {\bf 37}, 10822 (1988). %experimental measurement of aniKepler spectrum to 3 digits \bibitem{HMWRW} A. Holle, J. Main, G. Wiebusch, H. Rottke and K.H. Welge, {\em Phys. Rev. Lett.} {\bf 61}, 971 (1988). %experimental spectrum of chaotic diagmanetic Hydrogen atom \bibitem{RVJ84} R.V. Jensen, ``Stochastic ionizationof surface-state electrons: Classical Theory'' {\em Phys. Rev. \bf A 30}, 386 (1984). % fl_refs.tex 1/5-90 from % Periodic Orbit theory for smooth flows % Perron-Frobenius refs: \bibitem{P-F} see for ex. Y. Oono and Y. Takahashi, {\em Prog. Theor. Phys. \bf 63}, 1804 (1980); % discusses Perron-Frobenius theory, gets heuristically an % estimate for the trace in terms of inverse cycle stabilities, % and claims (wrongly) that Artin-Mazur-Ruelle zeta function % is inverse of the Fredholm determinant. % They say that instability of trajectories contributes % to stability of statistical properties in presence of weak noise \bibitem{CW81} S.-J. Chang and J. Wright, {\em Phys. Rev. \bf A 23}, 1419 (1981). \bibitem{TO84} Y. Takahashi and Y. Oono, {\em Prog. Theor. Phys. \bf 71}, 851 (1984). % complex paths, point group symmetry \bibitem{RCL} J.M. Robbins, S.C. Creagh and R.G. Littlejohn, {\em Phys. Rev. \bf A39}, 2838 (1989); %uniform quantization conditions {\em Phys. Rev. \bf A41}, 6052 (1990). \bibitem{CRL} S.C. Creagh, J.M. Robbins and R.G. Littlejohn, {\em Phys. Rev. \bf A42}, 1907 (1990). \bibitem{Creagh91} S.C. Creagh and R.G. Littlejohn, ``Semiclassical Trace Formulas in the Presence of Continuous Symmetries'', {\em Phys. Rev. \bf A 44}, 836 (1991). \bibitem{Creagh92} S.C. Creagh and R.G. Littlejohn, ``Semiclassical trace formulae for systems with non-Abelian symmetry'', {\em J. Phys. \bf A 25}, 1643 (1992). % 1643-1669 \bibitem{mcke} see for example H.P. McKean, %Selberg's Trace Formula, a nice discussion {\em Comm. Pure and Appl. Math. \bf 25 }, 225 (1972); {\bf 27}, 134 (1974). #the correction \bibitem{terras} see for ex. A. Terras, {\em Harmonic Analysis on Symmetric Spaces and Applications I} (Springer, Berlin 1985); H.P. McKean, %Selberg trace formula as applied to a compact Riemann surface {\em Comm. Pure \& Appl. Math., \bf 25}, 225 (1972); {\bf 27}, 134 (1974). \bibitem{mil72} W.H. Miller, {\em J. Chem. Phys. \bf56\rm, 38 (1972)} % early discussion of qm spectra extracted from isolated % stable or unstable period trajectories (old - but perhaps % Jacobian discussion useful \bibitem{miller} W.M. Miller, {\em Adv. Chem. Phys. \bf30\rm, 77 (1975).} % complex paths for 2-well potential \bibitem{Miller} W.M. Miller, \JCP{63}, 996 (1975). \bibitem{milcplx} W.H. Miller, {\em J. Phys. Chem. \bf83\rm, 960 (1979).} % complex paths, point group symmetry %polynomial 1-d Schroedinger eq. \bibitem{2} Y. Sibuya, {\it Global Theory of a Second Order Linear Ordinary Differential Operator with a Polynomial Coefficient\/}, North-Holland, Amsterdam (1975). \bibitem{2} C.M. Bender, K. Olaussen and P.S. Wang, Phys. Rev. {\bf D16} (1977) 1740-1748. \bibitem{3} R. Balian, G. Parisi and A. Voros, Phys. Rev. Lett. {\bf 41} (1978) 1141-1144, and in: {\it Feynman Path Integrals\/} (Proceedings, Marseille 1978), eds. S. Albeverio {\it et al.\/}, Lecture Notes in Physics {\bf 106}, Springer, Berlin (1979) p.337-360. \bibitem{4} G. Parisi, in: {\it The Riemann Problem, Complete Integrability and Arithmetic Applications\/}, eds. D. Chudnovsky and G. Chudnovsky, Lecture Notes in Mathematics {\bf 925}, Springer, Berlin (1982) p.178-183. \bibitem{5} F.T. Hioe, D. MacMillen and E.W. Montroll, J. Math. Phys. {\bf 17} (1976) 1320-1337. \bibitem{6} C.E. Reid, J. Mol. Spectrosc. {\bf 36} (1970) 183-191. \bibitem{10} J. Ecalle, {\it Cinq Applications des Fonctions R\'esurgentes\/} (chap.~1), Orsay Math. preprint 84T62 (1984, unpublished); E. Delabaere, H. Dillinger and F. Pham, Ann. Inst. Fourier {\bf 43} (1993) 163-199. %%%%%%%%%%%%%%%%%%%%%% QM ZETAS FINISHED %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% DYNAMO %%%%%%%%%%%%%%%%%%%%%% \bibitem{AG93} E.~Aurell and A.~Gilbert, % preprint 1992 {\em Geophys. \& Astrophys. Fluid Dynamics} ??? (1993). % Stephen Childress and Andrew D.Gilbert \bibitem{CGdynamo} S. Childress and A.D.~Gilbert {\em Stretch, Twist, Fold: The Fast Dynamo}, {\em Lecture Notes in Physics \bf 37} (Springer Verlag, Berlin 1995). \bibitem{} V. Oseledec, {\em Geophys. \& Astrophys. Fluid Dynamics} (1993), in press. \bibitem{BIS93} N.J. Balmforth, G.R. Ierley and E.A. Spiegel, Columbia Univ. preprint (1993); Submitted to {\em SIAM J. Applied Math.}. \bibitem{FDKO} J.M. Finn, J.D. Hanson, I. Kan and E. Ott, %``Do steady fast magnetic dynamos exist?'' {\em Phys. Rev. Lett. \bf 62, \rm 2965 (1989).} \bibitem{FDKO91} J.M. Finn, J.D. Hanson, I. Kan and E. Ott, %``Steady fast magnetic flows'' {\em Phys. Fluids \bf B 3, \rm 1250 (1991).} \bibitem{DO93a} Y. Du and E. Ott, %``Growth rates for fast kinematic dynamo instabilities of chaotic % fluid flows'' {\em J. Fluid Mech. \bf 257, \rm 265 (1993).} \bibitem{DO93b} Y. Du and E. Ott, %``Fractal dimensions of fast dynamo magnetic flows'' {\em Physica \bf D 67, \rm 387 (1993).} \bibitem{} Taylor, G.I. 1938. Proc. R. Soc. London {\bf A164:} 15. \bibitem{} Arneodo, A., Coullet, P. and Spiegel, E.A. 1985 Geophys. Astrophys. Fluid Dynamics {\bf 31:} 1. \bibitem{} Tresser, C. 1984. Ann. Inst. H. Poincar\'e {\bf 40:} 441. %%%%%%%%%%%%%%%%%%%%%% DYNAMO FINISHED %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% GOZZI-ana %%%%%%%%%%%%%%%%%%%%%% %Ennio Gozzi %3 nov 1992 %The lie-derivative is formula 20, and how it comes from %the Dirac deltas is briefly explained in the formulas which lead %to it or in E. Gozzi, et al. Phys. Rev. D 40, 3363 (1989). %The reason we get connection to the Liaupunov exponents is because %this formalism is similar to the Stochastic one of \bibitem{benz85} Benzi et al., {\em Jour. Phys. \bf A 18}, 2157 (1985). \bibitem{graham88} R. Graham, {\em Europhys. Lett. \bf 5}, 101 (1988). Graham and Tel, Phys. Rev. A 35, 1382 (1987) % Fogedby: Fokker-Planck plus kaos. Graham forudgriber % nogle af de ting jeg har arbejdet med, men forsoeger ikke % at lave feltteori. % there exists a special % class of generalized Lyapunov exponents which acquires a % natural interpretation in terms of this supersymmetric field theory. \bibitem{BPPV} R. Benzi, G. Paladin, G. Parisi and A. Vulpiani, {\em J. Phys. \bf A 18}, 2157 (1985). % the operator approach to CM pioneered by B.O. Koopman, {\em Proc. Nat. Acad. Sci. USA \bf 17}, 315 (1931); J.von Neumann, Ann.Math. 33, 587 (1932). % the Koopman operator and its inverse % the Perron-Frobenius operator \bibitem{LM85} A. Lasota and M. MacKey, {\em Probabilistic properties of deterministic systems} (Cambridge Univ. Press, 1985, Cambridge UK). %superceeded by: \bibitem{LM94} A. Lasota and M. MacKey, {\em Chaos, Fractals, and Noise; Stochastic Aspects of Dynamics} (Springer, Berlin 1994) T. Barnes and G.I.Ghandour, Nucl.Phys.B146, 483 (1978); V.I. Oseledec, Trans. Moscow Math.Soc. 19, 197 (1968) % "multiplicative cocycle": \bibitem{Reichl94} L.E. Reichl, {\em The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations} (Springer-Verlag, New York, 1992). %about topological pressure, variational principle for noncompact sets, Ya.B. Pesin, {\em Func. Anal. Applic. \bf 8}, 263 (1974). {A.M.Lyapunov, {\it "General problem of stability of motion"}, Ann.Of Math.Studies no.17, Princeton Univ.Press}. the concept of Lyapunov exponents generalized to stochastic systems {L.Arnold and V.Wihstutz,~"{\it Lyapunov exponents"}, Lecture Notes in Math. Vol 1186, Springer-Verlag, New York, 1986}. {M.V.Feigelman and A.M.Tsvelik, Sov.Phys.JETP {\bf 56} (1982) 823.; G.Parisi and N.Sourlas, Nucl.Phys.{\bf B206} (1982) 321; F.Langouche et al.{\it "Functional integration and semiclassical expansion}", Reidel, Dordrecht, 1982} that stochastic systems with a Langevin dynamics can be formulated via path-integrals and, once the noise is integrated away, they are equivalent to a one-dimensional supersymmetric field theory. \bibitem{GRT89} E. Gozzi, M. Reuter and W.D. Thacker, {\em Phys. Rev. \bf D 40}, 3363 (1989). E.Gozzi, M.Reuter, Phys.Lett.233B (1989) 383;E.238B (1990) 451; E.Gozzi, M.Reuter, Phys.Lett. 240B (1990) 137; \bibitem{GRT} E. Gozzi, M. Reuter and W.D. Thacker, {\em Phys. Rev. \bf D 46}, 757 (1992). E.Gozzi, M.Reuter and W.D.Thacker {\em Chaos, Solitons and Fractals \bf 2}, 441 (1992). {T. Barnes and G.I.Ghandour, Nucl.Phys. B146 (1978) 483; R.J.Rivers,"{\it Path-integral methods in quantum field theory}" Cambridge University Press, Cambridge, 1987} evaluation of the path-integrals {R.G.Littlejohn, Phys.Rep. 138 (1986) 193} lies in the lie-algebra of Sp(2n).}, {E.Witten, Nucl.Phys. B188 (1981) 513; P.Salomonson and J.W. Van Holten, Nucl. Phys. B196 (1982) 509; H.Nicolai, Phys.Lett. 89 B (1980)341} topological field theory E.Witten, Com.Math.Phys.117 (1988) 353; ibid. 118 (1988) 130; D.Birmingham et al.,~ Phys.Rep.209 (1991) 130}. \bibitem{BGS80} G. Bennettin, L. Galgani and J.-M. Strelcyn, % ``Lyapunove characteristic exponents for smooth % dynamical systems and for Hamiltonian systems: % a method for computing all of them", {\em Meccanica \bf 15}, 9 (1980). P.Salomonson and J.W Van Holten, Nucl. Phys. B181, 513 (1982) %%%%%%%%%%%%%%%%%%%%%% GOZZI-ana FINISHED %%%%%%%%%%%%%%%%%%%%%% %Pinball chaos - Monkey saddle: \bibitem{rod} D.L. Rod, {\em J. Diff. Equ.} {\bf 14}, 129 (1973). \bibitem{CPR75} R.C. 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Rev. \bf A 1992, Vol 45, Iss 12, pp 8530-8535 \bibitem{#} Grobgeld, D Pollak, E Zakrzewski, J %``A Numerical-Method for Locating Stable Periodic-Orbits in Chaotic Systems {\em Physica \bf D 1992, Vol 56, Iss 4, pp 368-380 \bibitem{#} Jung, C %``A Simple-Model System for Irregular Scattering {\em ACTA Physica \bf POLONICA B 1992, Vol 23, Iss 3, pp 177-218 \bibitem{#} Zyczkowski, K %``Classical and Quantum Billiards - Integrable, Nonintegrable and Pseudo-Integrable {\em ACTA Physica \bf POLONICA B 1992, Vol 23, Iss 3, pp 245-270 \bibitem{#} Banerjee, PP Banerjee, U Kaplan, H %``Response of an Acoustooptic Device with Feedback to Time- Varying Inputs {\em APPLIED OPTICS 1992, Vol 31, Iss 11, pp 1842-1852 \bibitem{#} Aurell, E Frisch, U Lutsko, J Vergassola, M %``On the Multifractal Properties of the Energy-Dissipation Derived from Turbulence Data {\em JOURNAL OF FLUID MECHANICS 1992, Vol 238, Iss MAY, pp 467-486 \bibitem{#} Grozdanov, TP Rakovic, MJ %``Periodic Orbit Theory of Broad Resonances in 2- Dimensional Hydrogenic Stark-Effect {\em J. Phys. \bf A 1991, Vol 24, Iss 23, pp 5517-5525 \bibitem{#} Jezewski, W %``Method of Characterization of Chaotic Trajectories {\em Phys. Lett. \bf A 1992, Vol 164, Iss 3-4, pp 274-278 \bibitem{#} Hasegawa, HH Saphir, WC %``Decaying Eigenstates for Simple Chaotic Systems {\em Phys. Lett. \bf A 1992, Vol 161, Iss 6, pp 471-476 \bibitem{#} Hasegawa, HH Saphir, WC %``Nonequilibrium Statistical-Mechanics of the Baker Map - Ruelle Resonances and Subdynamics {\em Phys. Lett. \bf A 1992, Vol 161, Iss 6, pp 477-482 \bibitem{#} Eckhardt, B Fishman, S Muller, K Wintgen, D %``Semiclassical Matrix-Elements from Periodic-Orbits {\em Phys. Rev. \bf A 1992, Vol 45, Iss 6, pp 3531-3539 \bibitem{#} Kovacs, Z Tel, T %``Bivariate Thermodynamics of Multifractals as an Eigenvalue Problem {\em Phys. Rev. \bf A 1992, Vol 45, Iss 4, pp 2270-2284 \bibitem{#} Mainieri, R %``Zeta-Function for the Lyapunov Exponent of a Product of Random Matrices {\em Phys. Rev. Lett. \bf 1992, Vol 68, Iss 13, pp 1965-1968 \bibitem{#} Tel, T %``Controlling Transient Chaos {\em J. Phys. \bf A 1991, Vol 24, Iss 23, pp 1359-1368 \bibitem{#} Gutowitz, HA %``Transients, Cycles, and Complexity in Cellular Automata {\em Phys. Rev. \bf A 1991, Vol 44, Iss 12, pp 7881-7884 \bibitem{#} Lewenkopf, CH Weidenmuller, HA %``Stochastic Versus Semiclassical Approach to Quantum Chaotic Scattering {\em ANNALS OF PHYSICS 1991, Vol 212, Iss 1, pp 53-83 \bibitem{#} Flepp, L Holzner, R Brun, E Finardi, M Badii, R %``Model Identification by Periodic-Orbit Analysis for NMR- Laser Chaos {\em Phys. Rev. Lett. \bf 1991, Vol 67, Iss 17, pp 2244-2247 \bibitem{#} Schaudt, KJ Kwong, NH Garcia, JD %``Exact-Solutions for Light-Scattering from Dielectric- Disk Arrays {\em Phys. Rev. \bf A 1991, Vol 44, Iss 6, pp 4076-4079 \bibitem{#} Dealmeida, AMO Saraceno, M %``Periodic Orbit Theory for the Quantized Bakers Map {\em ANNALS OF PHYSICS 1991, Vol 210, Iss 1, pp 1-15 \bibitem{#} Stoop, R Parisi, J Brauchli, H %``Convergence Properties for the Evaluation of Invariants from Finite Symbolic Substrings {\em HELVETICA Physica \bf ACTA 1991, Vol 64, Iss 6, pp 950-951 \bibitem{#} Stoop, R %``Phase-Transitions in the Generalized Entropy Spectrum of Nonhyperbolic Dynamic-Systems {\em ZEITSCHRIFT FUR NATURFORSCHUNG SECTION A-A JOURNAL OF Physica \bfL SCIENCES 1991, Vol 46, Iss 12, pp 1117-1122 \bibitem{Stoop97} Stoop R, Steeb WH ``Chaotic family with smooth Lyapunov dependence'', {\em Phys. Rev. \bf E 55}, 7763 (1997). \bibitem{#} Jung, C Tel, T %``Dimension and Escape Rate of Chaotic Scattering from Classical and Semiclassical Cross-Section Data {\em J. Phys. \bf A 1991, Vol 24, Iss 12, pp 2793-2805 \bibitem{#} Robbins, JM %``Maslov Indexes in the Gutzwiller Trace Formula {\em Nonlinearity \bf 1991, Vol 4, Iss 2, pp 343-363 \bibitem{#} Rechester, AB White, RB %``Symbolic Kinetic-Equation for a Chaotic Attractor {\em Phys. Lett. \bf A 1991, Vol 156, Iss 7-8, pp 419-424 \bibitem{#} Dmitriev, AS Panas, AI Starkov, SO %``Storing and Recognizing Information Based on Stable Cycles of One-Dimensional Maps {\em Phys. Lett. \bf A 1991, Vol 155, Iss 8-9, pp 494-499 \bibitem{#} Defilippo, S Salerno, M %``On a Procedure to Evaluate Unstable Periodic-Orbits {\em Phys. Lett. \bf A 1991, Vol 153, Iss 4-5, pp 173-176 \bibitem{#} Tel, T %``Thermodynamics of Chaotic Scattering at Abrupt Bifurcations {\em Phys. Rev. \bf A 1991, Vol 44, Iss 2, pp 1034-1043 \bibitem{#} Prakash, S Peng, CK Alstrom, P %``Deterministic Diffusion Generated by a Chaotic Map {\em Phys. Rev. \bf A 1991, Vol 43, Iss 12, pp 6564-6571 \bibitem{#} Kus, M Zakrzewski, J Zyczkowski, K %``Quantum Scars on a Sphere {\em Phys. Rev. \bf A 1991, Vol 43, Iss 8, pp 4244-4248 \bibitem{#} Pawelzik, K Schuster, HG %``Unstable Periodic-Orbits and Prediction {\em Phys. Rev. \bf A 1991, Vol 43, Iss 4, pp 1808-1812 \bibitem{#} Braiman, Y Goldhirsch, I %``Taming Chaotic Dynamics with Weak Periodic Perturbations {\em Phys. Rev. Lett. \bf 1991, Vol 66, Iss 20, pp 2545-2548 \bibitem{#} Fujisaka, H Shibata, H %``New Asymptotic Law Governing Overall Temporal Correlations in Chaotic Systems - Ensemble Homogenization and Order-Q Power Spectrum {\em PROGRESS OF THEORETICAL PHYSICS 1991, Vol 85, Iss 2, pp 187-204 \bibitem{#} Legrand, O Sornette, D %``1st Return, Transient Chaos, and Decay in Chaotic Systems {\em Phys. Rev. Lett. \bf 1991, Vol 66, Iss 16, pp 2172-2172 \bibitem{#} Dmitriyev, AS %``Recording and Recognition of Patterns in One-Dimensional Dynamic-Systems {\em RADIOTEKHNIKA I ELEKTRONIKA 1991, Vol 36, Iss 1, pp 101-108 %%%%%%%%%%%%%%%%%%%%%% RONNIE-ana FINISHED %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% BRAIDS %%%%%%%%%%%%%%%%%%%%%% % Braid analysis of (low-dimensional) chaos} % T.D.H. Hall, N.B. Tufillaro % 21 April 1993 \bibitem{art1} E.\ Artin, American Scientist {\bf 38}, 112 (1950). \bibitem{boy1} P.\ Boyland, ``Braid types and a topological method of proving positive entropy'' (1984, unpublished); Contemp.\ Math.\ {\bf 81}, 119 (1988). \bibitem{mat1} T.\ Matsuoka, in {\it Dynamical systems and nonlinear oscillations}, edited by G.\ Ikegami (World Scientific, Singapore, 1986) p.\ 58; Invent.\ Math.\ {\bf 70}, 319 (1983); J.\ Diff.\ Eqs.\ {\bf 76}, 190 (1988). \bibitem{mel1} P.\ Melvin and N.\ B.\ Tufillaro, Phys.\ Rev.\ A {\bf 44}, R3419 (1991), and references therein. \bibitem{bald} S.\ Baldwin, Discrete Math.\ {\bf 67}, 111 (1987); C.\ Bernhardt, E.\ Coven, M.\ Misiurewicz, and I.\ Mulvey, Trans.\ Am.\ Math.\ Soc.\ {\bf 333}, 701 (1992). \bibitem{bes} M.\ Bestvina and M.\ Handel, Ann.\ Math.\ {\bf 135}, 1 (1992); ``Train tracks for surface homeomorphisms,'' (1992, unpublished). \bibitem{los} J.\ Los, ``Pseudo-Anosov maps and invariant train tracks in the disc: a finite algorithm,'' Proc.\ Lond.\ Math.\ Soc.\ (1993, to appear). \bibitem{franks} J.\ Franks and M.\ Misiurewicz, ``Cycles for disk homeomorphisms and thick trees,'' (unpublished). \bibitem{nbtrel} N.\ B.\ Tufillaro, H.\ Solari, and R.\ Gilmore, Phys.\ Rev.\ A {\bf 41}, 5717 (1990). \bibitem{li} T.\ Li and J.\ Yorke, Amer.\ Math.\ Monthly {\bf 82}, 985 (1975). \bibitem{gam1} J.\ M.\ Gambaudo, S.\ van Strien, and C.\ Tresser, Ann.\ Inst.\ H.\ Poincare Phys.\ Theor.\ {\bf 50}, 335 (1989). \bibitem{mind1} G.\ B.\ Mindlin, H.\ Solari, M.\ Natiello, R.\ Gilmore, and X-J.\ Hou, J.\ Nonlinear Sci.\ {\bf 1}, 146 (1991); G.\ B.\ Mindlin and R.\ Gilmore, Physica D {\bf 58}, 229 (1992). \bibitem{toby} T.\ D.\ H.\ Hall, ``Weak universality in two-dimensional transitions to chaos,'' (1992, submitted to Phys.\ Rev.\ Lett.); ``Periodicity in chaos: the dynamics of surface automorphisms,'' Ph.D. Thesis, University of Cambridge (1991, unpublished). \bibitem{nit} M.\ Misiurewicz and Z.\ Nitecki, {\it Combinatorial Patterns for Maps of the Interval}, Mem.\ Am.\ Math.\ Soc.\ {\bf 456} (1991). \bibitem{thur} W.\ Thurston, Bull.\ Amer.\ Math.\ Soc.\ (N.S.) {\bf 19}, 417 (1988). \bibitem{hand} M.\ Handel, Comm.\ Math.\ Phys.\ {\bf 127}, 339 (1990); R.\ C.\ Penner and J.\ L.\ Harer, {\it Combinatorics of Train Tracks}, Ann.\ Math.\ Studies {\bf 125} (Princeton Univ.\ Press, Princeton, 1992). \bibitem{boy2} P.\ Boyland, Comment.\ Math.\ Helv.\ (accepted for publication). \bibitem{tad} T.\ White, (unpublished). A copy of the train track program is available by contacting T. White at ``tadpole@ucrmath.ucr.edu''. \bibitem{block} See Lemma 6.6 of Ref.\ \cite{toby}; L.\ Block, J.\ Guckenheimer, M.\ Misiurewics, and L-S.\ Young, in {\it Global Theory of dynamical Systems}, Lecture Notes in Mat.\ vol.\ 819 (Springer-Verlag, New York, 1980), p.\ 18; program is available from first author. \bibitem{sch} K. Pawelzik and H.\ G.\ Schuster, Phys.\ Rev.\ A {\bf 43}, 1808 (1991). \bibitem{mind2} G.\ B.\ Mindlin, X-J.\ Hou, H.\ G.\ Solari, R.\ Gilmore, and N.\ B.\ Tufillaro, Phys.\ Rev.\ Lett.\ {\bf 64}, 2350 (1990); F. Papoff, E.\ Arimondo, F.\ Fioretto, G.\ B.\ Mindlin, H.\ G.\ Solari, and R.\ Gilmore, Phys.\ Rev.\ Lett.\ {\bf 68}, 1128 (1992). \bibitem{nbtnmr} N.\ B.\ Tufillaro, R.\ Holzner, L.\ Flepp, E.\ Brun, M.\ Finardi, and R.\ Badii, %``Template Analysis for a Chaotic NMR Laser {\em Phys.\ Rev.\ \bf A 44}, R4786 (1991). \bibitem{mor} E.\ A.\ Elrifai and H.\ R.\ Morton, ``Algorithms for positive braids,'' (1990, unpublished). \bibitem{jaq} A.\ Jaquemard, J.\ Pure Appl.\ Algebra {\bf 63}, 161 (1990). \bibitem{bz} A. Arnedo, F.\ Argol, J.\ Elezgaray, and P. Richetti, Physica D {\bf 62}, 134 (1993); D.\ P.\ Lathrop and E.\ J.\ Kostelich, Phys.\ Rev.\ A {\bf 40}, 4028 (1989); K. Coffman, W.\ D.\ McCormick, A.\ Noszticzius, R.\ H.\ Simoyi, and H.\ Swinney, J. Chem.\ Phys {\bf 86}, 119 (1987); J.\ C.\ Roux, R.\ H.\ Simoyi, and H.\ L.\ Swinney, Physica D {\bf 8}, 257 (1983). %%%%%%%%%%%%%%%%%%%%%% BRAIDS FINISHED %%%%%%%%%%%%%%%%%%%%%% \bibitem{6.} J.D. Farmer and J.J. Sidorowich, {\em Phys. Rev. Lett. \bf 59}, 845 (1987); J.P. Crutchfield and B.S. McNamara, {\em Complex Systems \bf 1}, 417 (1987); E.J. Kostelich and J.A. Yorke {\em Phys. Rev. \bf A38}, (1988). %%%%%%%%%%%%%%%%%%%%%% CRACKS %%%%%%%%%%%%%%%%%%%%%% \bibitem{BDS89} M. Barber, J. Donley and J.S. Langer %``Steady-state propagation of a crack in a viscoelastic strip'' {\em Phys. Rev. \bf A 40}, 366 (1989). \bibitem{L92} J.S. Langer %``Models of crack propagation'' {\em Phys. Rev. \bf A 46}, 3123 (1992). \bibitem{L93} J.S. Langer %``Models of crack propagation. II. Two-dimensional model with % dissipation on the fracture surface'' {\em Phys. Rev. \bf A 48}, 439 (1993). \bibitem{CLS93} J.S. Langer % review of the field {\em Rev. Mod. Phys. (Colloquium) \bf ?? }, (already out?) (1994). (some more coming up, but preprint unfinished - can try langer@itp.ucsb.edu, if right address) %%%%%%%%%%%%%%%%%%%%%% CRACKS FINISHED %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%RANDOM NUM. GENERATION %%%%%%%%%%%%%%%%%%%%%%%%% % Kolmogorov-Anosov K systems, nonzero entropy \bibitem{kolmogorov}A.N.Kolmogorov, Dokl.Akad.Nauk SSSR 119, 861 (1958) [Russian]. % the Kolmogoroff-Sinai (KS) entropy M.Casartelli et al., Phys.Rev.A 13 (1921) 1976; G.Benettin et al., Phys.Rev.A 14 (2338) 1976}. % nonzero entropy savvidy2,savvidy3}. \bibitem{savvidy1}G.K.Savvidy,N.G.Ter-Arutyunian-Savvidy, On the Monte-Carlo Simulation of Physical Systems, Preprint EPI-865(16)-86, Yerevan Jun.1986 [Russian]; J.Comput.Phys. 97, 566 (1991). \bibitem{savvidy2}G.K.Savvidy, Nucl.Phys.B246, 302 (1984). \bibitem{savvidy3}G.K.Savvidy.Phys.Lett.130B, 303 (1983). % matrix generator for random numbers: \cite{savvidy1,akopov1} \bibitem{akopov1}N.Z.Akopov,G.K.Savvidy,N.G.Ter-Arutyunian-Savvidy, Matrix Generator of Pseudorandom Numbers, Preprint EPI-867(18)-86, Yerevan Jun.1986 [Russian]; J.Comput.Phys.97, 573 (1991). \bibitem{akopov2}N.Z.Akopov, E.M.Madunts, G.K.Savvidy, A new matrix generator for lattice simulation, in Proceedings of Computing in High Energy Physics`91 International Conference, pp.477-479 (Tsukuba, Japan, 1991). \bibitem{abramyan}R.O.Abramyan, N.Z.Akopov, G.K.Savvidy, N.G.Ter-Arutyunian-Savvidy, Sinai Billiards as a Pseudorandom Number Generator, Preprint EPI-922(73)- 86, Yerevan 1986 [Russian]; G.A.Galperin, N.I.Chernov.Billiardi i Chaos. Matematika i Kibernetika 5, 10 (1991), (Znanie, Moskva, 1991),[Russian]. \bibitem{AMNSG} N.Z.Akopov,E.M.Madounts, A.B.Nersesian, G.K.Savvidy and W. Greiner % ``Fast K system generator of pseudorandom numbers}'' %From savvidy@knosos.cc.uch.gr Nov 20 1993 \bibitem{niederreiter}H.Niederreiter. Math. Japonica 31, 759 (1986). \bibitem{grothe}H.Grothe. Zufallszahlen und Simulation (Teubner, Stuttgard, 1986); Statist. Papers 28, 233 (1987). %%%%%%%%%%%%%%%%%%%%%RANDOM NUM. GENERATION FINISHED%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%NUMERICAL METHODS %%%%%%%%%%%%%%%%%% \bibitem{Earn} D.J.D Earn, %David Earn {\em Symplectic integration without roundoff error}, astro-ph/9408024. % seems a good reference for literature etc.: \bibitem{Earn} B.A. Shadwick, J.C. Bowman, and P.J. Morrison, %(Institute for Fusion Studies, % The University of Texas at Austin, Austin, TX 78712, USA) {\em Exactly Conservative Integrators}, chao-dyn/9507012, % [on preprint shelf) Submitted to SIAM J. Sci. Comput. %%%%%%%%%%%%%%%%%%%%%NUMERICAL METHODS FINISHED %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%HOLOMORPHIC DYNAMICS %%%%%%%%%%%%%%%%%% \bibitem{Levin89} A.\'E. Eremenko and G.M. Levin, `Periodic points of polynomials', {\em Ukrain. Mat. J. \bf 41}, 1258 (1989), [english translation]. \bibitem{Levin91} G. Levin, M. Sodin and P. Yuditskii, `A Ruelle operator for a real Julia set', {\em Comm. Math. Phys.} {\bf 141}, 119 (1991). % pp. 119--131 \bibitem{Levin92} G. Levin, M. Sodin, and P. Yuditskii, `Ruelle operators with rational weights for Julia sets', {\em J. d'Analyse Math. 63}, 303 (1994). \bibitem{Levin93} A. Eremenko, G. Levin, and M. Sodin, `On the distribution of zeros of a Ruelle zeta-function', {\em Comm. Math. Phys.} {\bf 159}, 433 (1994). \bibitem{Levin94} G. Levin, `On Mayer's conjecture and zeros of entire functions', {\em Ergod. The. and Dynam Sys. \bf 14}, 565 (1994). % DLA-Julia set potential is computed on the Julia set % looks similar to the Levin stuff % A sum for P.O.-s leads to a sum for images of the critical point \bibitem{Zeitak88} I. Procaccia and R. Zeitak, % ``Shape of Fractal Growth Patterns: ...'', {\em Phys. Rev. Lett. \bf 60}, 2511 (1988). \bibitem{WBKS} M. Widom, D. Bensimon, L.P. Kadanoff and S.J. Shenker, {\em Strange objects in the complex plane}, {\em J. Stat. Phys. {\bf 32}}, 443 (1983). \bibitem{Eckh93} B. Eckhardt, ``Escape rates and Hausdorff dimension of Julia sets'', {\em Chaos, Solitons and Fractals \bf 3}, 387 (1993). \bibitem{Ander_95} G. Anderson, {\em Notes on dynamical zeta functions}, (U. of Minessota preprint, March 1995). \bibitem{KHans95} Kim Hansen, {\em Bachelorprojekt} (Niels Bohr Insitute, May 1995). \bibitem{fatou} P. Fatou, {\em Bull. Soc. Math. France \bf 47}, 161 (1919); ibid. {\bf 48}, 33 and 208 (1920). \bibitem{julia} G. Julia, {\em J. Math. Pures et Appl. \bf 4}, 47 (1918). %%%%%%%%%%%%%%%%%%%%%HOLOMORPHIC DYNAMICS FINISHED %%%%%%%%%%%%%%%%%%