%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. Ketoja and Juhani Kurkij\"arvi, ``BINARY TREE APPROACH TO SCALING IN UNIMODAL MAPS'' \bibitem{4} M. Misiurewicz, Inst. Hautes \'Etudes Sci. Publ. Math. 53, 17 (1981); for the most recent developments see G. Keller and T. Nowicki, Commun. Math. Phys. 149, 31 (1992), and references therein. \bibitem{7} J.A. Ketoja and J. Kurkij\"arvi, Phys. Rev. A 33, 2846 (1986). \bibitem{8} J.D. Farmer, Phys. Rev. Lett. 55, 351 (1985); C. Grebogi, S.W. McDonald, E. Ott, and J.A. Yorke, Phys. Lett. A 110, 1 (1985); G. Gao and G. Hu, Commun. Theor. Phys. 10, 127 (1988). \bibitem{9} J.-P. Eckmann, H. Epstein, and P. Wittwer, Commun. Math. Phys. 93, 495 (1984); R. Delbourgo and B.G. Kenny, Phys. Rev. A 33, 3292 (1986). \bibitem{11} T. Post and H.W. Capel, Physica A 178, 62 (1991). \bibitem{12} J. Dias de Deus, R. Dil\~ao, and A. Noronha da Costa, Phys. Lett. A 101, 459 (1984). \bibitem{13} K. Shibayama, in: The theory of dynamical systems and its applications to nonlinear problems, ed. H. Kawakami (World Scientific, Singapore, 1984) p. 124. \bibitem{14} J.A. Ketoja and O.-P. Piiril\"a, Phys. Lett. A 138, 488 (1989). \bibitem{15} M. Lyubich and J. Milnor, The Fibonacci unimodal map. Preprint IMS91-15. \bibitem{16} Y. Ge, E. Rusjan, and P. Zweifel, J. Stat. Phys. 59, 1265 (1990). \bibitem{17} B. Derrida, A. Gervois, and Y. Pomeau, J. Phys. A 12, 269 (1979). \bibitem{19} M. Lyubich, A talk at the Workshop on Renormalisation in Dynamical Systems, University of Warwick, Coventry, 30 March 1992. \bibitem{21} H.-O. Peitgen and P.H. Richter, The Beauty of Fractals (Springer, Berlin, 1986). \bibitem{daniels_vallieres_yuan}V. Daniels, M. Valli\`eres and J-M. Yuan, % Chaotic scattering on a double well: Periodic orbits, symbolic % dynamics, and scaling. {\em Chaos}, {\bf 3}, 475, (1993). %%%%%%%%%%%%%%%%%%%%%% CLASSICAL DYNAMICAL SYSTEMS FINISHED %%%%%%%%%%%%%% %%%%%%%%%%%%% BALADI %%%%%%%%%%%%%%%%%%%%%% % baladi@umpa.ens-lyon.fr Dec 28, 92, from \bibitem{BY} V. Baladi and L.-S. Young %``On the spectra of randomly perturbed expanding maps submitted to {\em Comm. Math. Phys.}. \bibitem{BK90} V. Baladi and G. Keller, {\em ``Zeta functions and transfer operators for piecewise monotone transformations''}, {\em Comm. Math. Phys. \bf 127}, 459 %--477 (1990). \bibitem{2Baladi} P. Baxendale, ``Brownian motions in the diffeomorphism group", {\em Compositio Math. \bf 53}, 19--50 (1984). \bibitem{3Baladi} M. Benedicks and L.-S. Young, ``Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps'', {\em Ergodic Theory \& Dynamical Systems \bf 12}, 13--37 (1992). \bibitem{4Baladi} P. Collet, ``Ergodic properties of some unimodal mappings of the interval'', Preprint Mittag-Leffler (1984). \bibitem{5 }same, % ``Some ergodic properties of maps of the interval , in {\em ``Dynamical Systems and Frustrated Systems \toappear \eds R. Bamon, J.-M. Gambaudo and S. Martinez (1991) \bibitem{7Baladi} E.M. Coven, I. Kan and J.A. Yorke, ``Pseudo-orbit shadowing in the family of tent maps'', {\em Trans. Amer. Math. Soc. \bf 308}, 227--241 (1988). \bibitem{6 } P. Collet and S. Isola, % ``On the essential spectrum of the transfer operator % for expanding Markov maps {\em Comm. Math. Phys. \bf 139 }, 551--557 (1991) \bibitem{8 } J. Franks, % ``Manifolds of $\CC^r$ mappings and applications % to differentiable dynamical systems {\em Studies in Analysis, Adv. Math. Suppl. Stud. \bf 4 }, 271--291 (1979) \bibitem{HK82} F. Hofbauer and G. Keller, {\em ``Ergodic properties of invariant measures for piecewise monotonic transformations''}, {\em Math. Z. \bf 180}, 119 %--140 (1982). \bibitem{HK84} F. Hofbauer and G. Keller, {\em ``Zeta-functions and transfer-operators for piecewise linear transformations''}, {\em J. reine angew. Math. \bf 352}, 100 %--113 (1984). \bibitem{10 } G. Keller, % ``Stochastic stability in some chaotic dynamical systems" {\em Monatsh. Math. \bf 94 }, 313--333 (1982) \bibitem{K84}G. Keller, {\em ``On the rate of convergence to equilibrium in one-dimensional systems''}, {\em Comm. Math. Phys. \bf 96}, 181 %--193 (1984). \bibitem{Keller98} G. Keller, {\em Equilibrium states in ergodic theory} (Cambridge Univ. Press, Cambridge 1989). \bibitem{12Baladi} Y. Kifer, ``On small random perturbations of some smooth dynamical systems", {\em Math. USSR-Izv. \bf 8}, 1083--1107 (1974). \bibitem{13Baladi} Y. Kifer, {\em Ergodic Theory of Random Transformations} (Birkh\"auser, Boston, Basel 1986). \bibitem{14Baladi} Y. Kifer, {\em Random Perturbations of Dynamical Systems} (Birkh\"auser, Boston 1988). \bibitem{15Baladi} Y. Kifer, ``A note on integrability of $C^r$ norms of stochastic flows and applications'', in {\em ``Stochastic Mechanics and Stochastic Processes, Proc. Conf. Swansea/UK 1986}, {\em Springer Lecture Notes in Math. \bf 1325}, 125--131 (Springer Verlag, Berlin 1988). \bibitem{B95} V. Baladi, {\em ``Infinite kneading matrices and weighted zeta functions of interval maps'}, {\em J. Functional Analysis \bf 128}, 226 %-244 (1995). \bibitem{24 } M. Rychlik, % ``Bounded variation and invariant measures" {\em Studia Math. \bf LXXVI }, 69--80 (1983) \bibitem{25 } J.H. Wilkinson, {\em The Algebraic Eigenvalue Problem (Oxford University Press , London 1965) \bibitem{26 } S. Wong, % ``Some metric properties of piecewise % monotonic mappings of the unit interval {\em Trans. Amer. Math. Soc. }, 493--500 (1978) %%%%%%%%%%%%% BALADI finshed %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% PERIODIC ORBITS EXTRACTION: %%%%%%%%%%%%%%%%%% % this paper is one of the premordial ``chaology'' papers and it (or another % Moore and Spiegel paper) might or % might not contain method for finding periodic orbits \bibitem{MS66} D.W. Moore and E.A. Spiegel, ``A thermally excited nonlinear oscillator'', {\em Astrophys. J., \bf 143}, 871 (1966). % the method for calculating peridoic orbits is laid out \bibitem{BMS71} N.H. Baker, D.W. Moore and E.A. Spiegel, {\em Quar. J. Mech. and Appl. Math. \bf 24}, 391 (1971). %read by blurb: \bibitem{EAS87} E.A. Spiegel, {\em Chaos: a mixed metaphor for turbulence}, {\em Proc. Roy. Soc. \bf A413}, 87 (1987). \bibitem{baranger} M. Baranger and K.T.R. Davies {\em Ann. Physics \bf 177}, 330 (1987). \bibitem{varcyc} B.D. Mestel and I. Percival, {\em Physica D} {\bf 24}, 172 (1987); Q. Chen, J.D. Meiss and I. Percival, {\em Physica D} {\bf 29}, 143 (1987). \bibitem{Helleman} find Helleman et all Fourier series methods % residue introduced here \bibitem{gree98} J.M. Greene, ``Two-Dimensional Measure-Preserving Mappings'', {\em J. Math. Phys. \bf 9\rm, 760 (1968)} % A particular area-preserving mapping of a plane onto itself has been studied in detail with the aid of a digital computer. A large number of fixed points, finite sets of points that transform into each other, were located and classified as elliptic or hyperbolic depending on the nature of the linearized mapping in the neighborhood. % A quantity called the residue was calculated for each fixed point. This quantity can be used to predict whether other nearby fixed points are elliptic or hyperbolic. % The results showed that there are considerable regions in which almost all the fixed points are hyperbolic. Further calculations were made to estimate the area enclosed by the invariant curves whose existence has been established by Moser. %refer to Green's symmetry lines \bibitem{gree} J.M. Greene, {\em J. Math. Phys. \bf 20\rm, 1183 (1979)} %Henon periodic orbit by variational minimization \bibitem{varhenon} O. Biham and W. Wenzel, ``Characterization of Unstable Periodic Orbits in Chaotic Attractors and Repellers'', {\em Phys. Rev. Lett. \bf 63}, 819 (1989). \bibitem{biham_wenzel_90} O. Biham and W. Wenzel, {\em Phys. Rev. A}{ \bf 42}, 4639 (1990). \bibitem{biham_wenzel_91} Wenzel, W Biham, O Jayaprakash, C %``Periodic-Orbits in the Dissipative Standard Map {\em Phys. Rev. \bf A 43}, 6550 (1991). % 6550-6557 \bibitem{NY} H.E. Nusse and J. Yorke, "A procedure for finding numerical trajectories on chaotic saddles" {\em Physica \bf D 36}, 137 (1989). \bibitem{GJP} G. Gunaratne, M.H. Jensen and I. Procaccia, {\em Nonlinearity \bf 1}, 157 (1988). \bibitem{LK} D.P. Lathrop and E.J. Kostelich, "Characterization of an experimental strange attractor by periodic orbits" \bibitem{HDB} T. E. Huston, K.T.R. Davies and M. Baranger {\em Chaos \bf 2}, 215 (1991). % periodic orbits in Henon-Heiles \bibitem{BBLM} M. Brack, R. K. Bhaduri, J. Law and M. V. N. Murthy, {\em Phys. Rev. Lett. \bf 70}, 568 (1993). % analytic formulas for periodic orbits in Henon-Heiles \bibitem{GIRST} Z. Gills, C. Iwata, R. Roy, I.B. Scwartz and I. Triandaf, ``Tracking Unstable Steady States: Extending the Stability Regime of a Multimode Laser System'', {\em Phys. Rev. Lett. \bf 69}, 3169 (1992). \bibitem{Moss94} F. Moss, {\em ``Chaos under control''}, {\em Nature \bf 370}, 615 (1994). \bibitem{Schiff94} S.J. Schiff, et al. {\em ``Controlling chaos in the brain''}, {\em Nature \bf 370}, 615 (1994). \bibitem{Glanz} J. Glanz, (FIND!), speculated applications of chaos to epilepsy and the brain, chaos-control, {\em Science \bf 265}, 1174 (1994). \bibitem{Glanz1} J. Glanz, {\em ``Mastering the Nonlinear Brain''}, {\em Science \bf 227}, 1758 (1997). % mentions my cycles as useful to these people {http://www.krasnow.gmu.edu/neuraldyn/papers/others/list.html} \bibitem{So97} P. So, E. Ott, T. Sauer, B.J. Gluckman, C. Grebogi and S.J. Schiff, {\em ``Extracting Unstable Periodic Orbits from Chaotic Time Series Data''}, {\em Phys. Rev. \bf E 55}, 5398 (1997). %5398-5417 \bibitem{So96} P. So, E. Ott, S. J. Schiff, D. T. Kaplan, T. Sauer and C. Grebogi, {\em ``Detecting Unstable Periodic Orbits in Chaotic Experimental Data}, {\em Phys. Rev. Lett. \bf 76}, 4705 (1996). \bibitem{diak} P. Schmelcher and F.K. Diakonos, ``Detecting Unstable Periodic Orbits of Chaotic Dynamical Systems'', Phys. Rev. Letts (1997) \bibitem{Isola90} S. Isola, %`$\zeta$-functions and distribution of periodic %orbits of toral automorphisms ', {\em Europhysics Letters \bf 11}, 517 (1990). %pp. 517--522 Zoldi SM Unstable periodic orbit analysis of histograms of chaotic time series PHYS REV LETT 81: (16) 3375-3378 OCT 19 1998 \bibitem{ZG96} S.M. Zoldi and H.S. Greenside, %``Spatially localized unstable periodic orbits % of a high-dimensional chaotic system'', {\em Phys. Rev. \bf E 57}, R2511 (1998). % R2511-R2514 % chao-dyn/9704005 S.M. Zoldi and H.S. 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. Giberti and Zhiming Zheng % ``Characterization of the Lorenz attactor by unstable periodic orbits", {\em Nonlinearity \bf 6}, 251 (1993). \bibitem{McN} Sean McNamara, % ``A periodic expansion of the Lorenz system'' {\em Geophysical Fluid Dynamics Summer School 1993 } % N.J. Balmforth's summer student - notes for summer project %%%%%%%%%%%%%%%%%%%%%%% LORENZ FINISHED %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% HENON %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bibitem{henon} M. H\`enon, {\em Comm. Math. Phys. \bf 50 }, 69 (1976). \bibitem{lozi2} R. Lozi, {\em J. Phys. (Paris) Colloq. \bf 39}, 9 (1978). \bibitem{Mira} C. Mira, {\em Chaotic Dynamics} (World Scientific, Singapore 1987) %Myrberg trees: \bibitem{fou1} D. Fournier, H. Kawakami and C. Mira, {\em C.R.Acad.Sci.Ser.I, \bf 301}, 325 (1985). %bifurcation locus in a-b plane \bibitem{fou} D. Fournier, H. Kawakami and C. Mira, ?cra 298, 253, 1984 %pictures of refoldings in b-a parameter plane : \bibitem{fou2} D. Fournier, H. Kawakami and C. Mira, ?cra 301, 223, 1985 %Myrberg trees \bibitem{kaw} H. Kawakami and C. Mira, {\em preprint Syst. Dyn. INSA 85-4} %Myrberg trees \bibitem{carc} J.P. Carcass\'es, %``Determination of different configurations of fold and flip % bifurcation curves of a one or two-dimensional map'' {\em Int. J. of Bifurcations and Chaos \bf 3}, 869 %-902 (1993). \bibitem{hol} P.J. Holmes and D. Whitney, {\em Phil. Trans. Roy. Soc. \bf A 311}, 43 (1984). \bibitem{hol2} P.J. Holmes, {\em Phys. Rev. Lett. \bf 104A}, 299 (1984). \bibitem{hol2} P.J. Holmes, \PLA{104}, 299 (1984). \bibitem{hol} P.J. Holmes and R.\ Williams, Arch.\ Rational Mech.\ Anal.\ {\bf 90}, 115 (1985). \bibitem{myr} P. J. Myrberg, {\em Ann. Acad. Sc. Fenn., Ser. \bf A 259}, 1 (1958) \bibitem{sa} A.\ Sarkovskii, {\em Ukrain.\ Mat.\ Z.\ \bf 16}, 61 (1964). \bibitem{MSS} N. Metropolis, M.L. Stein and P.R. Stein, {\em J. Comb. Theo. } {\bf A15}, 25 (1973) % henon primary tangs; best estimate of the entropy \bibitem{gkantz} P. Grassberger and H. Kantz, {\em Phys. Lett. A} {\bf 113}, 235 (1985). \bibitem{kantzg} H. Kantz and P. Grassberger, {\em Physica } {\bf 17D}, 75 (1985). \bibitem{GKM} P. Grassberger, H. Kantz and U. Moening, {\em J. Phys. \bf A 43}, 5217 (1989). \bibitem{front} P. Cvitanovi\'c, in preparation. Some numerical evidence for the correctness of the pruning front conjecture is given in refs.~\cite{CGP,GKM}. \bibitem{AGIP} G. D'Alessandro, P. Grassberger, S. Isola and A. Politi, % On the topology of the Henon Map {\em J. Phys. \bf A 23}, 5285 (1990). % ISI Torino preprint (Oct 1990) \bibitem{#} G. D'Alessandro, S. Isola, A. 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. Phys. \bf 49}, 157 (1982). \bibitem{mizs} Misziurowicz on Lozi \bibitem{J81} M. Jakobson, {\em Commun. Math. Phys. \bf 81}, 39 (1981). \bibitem{BC85} M. Benedicks and L. Carleson, {\em Ann. of Math.}, {\bf 122}, 1 (1985). \bibitem{BC89} M. Benedicks and L. Carleson, {\em IXth Int. Congr. on Mathematical Physics}, B. Simon {\em et al.}, eds., p.489, (Adam Hilger, Bristol, 1989). % widely advertised preprint from the Royal Instute of % Technology, Stockholm (1988) about the H\'enon map. \bibitem{BC91} M. Bennedicks and L. Carleson, % The dynamics of the H\'enon map {\em Ann. of Math. \bf 133}, 73 (1991). \bibitem{McRob} F.A.~McRobie, %``Bifurcational precedences in the braids of periodic %orbits of spiral 3-shoes in driven oscillators" {\em Proc. R. Soc. Lond. \bf A 438}, 545 (1992?) %545-569 % about bifurcation and symbolic dynamics in two dimensional maps. \bibitem{} Shil'nikov, L.P. 1965. Soc. Math. Dokl. {\bf 6:} 163. Shil'nikov, . 1970. Math. USSR Sbornik {\bf 10:} 91. \bibitem{GST} S.V. Gonchenko, L.P. Shil'nikov and D.V. Turaev % met Turaev at Woods Hole '93 - smart young man %% one might make not only the first derivatives coincide % at homoclinic tangency, but align arbitrarily many derivatives. % ``On models with non-rough Poincar\'e homoclinic curves'' {\em Physica \bf D 62}, 1 (1993). \bibitem{YI94} Y. Ishii, ``Towards the Kneading Theory for Lozi Attractors. I. Critical Sets and Pruning Fronts'', Kyoto Univ. Math. Dept. preprint (Feb. 1994). \bibitem{YI95} Y. Ishii, ``Towards the Kneading Theory for Lozi Attractors. II. A solution of the pruning front conjecture and partial monotonicity of the topological entropy'', Univ. Paris-Sud preprint (Apr. 1994). \bibitem{YI96} Y. Ishii, ``Towards a kneading theory for Lozi mappings. I. 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. Phys.\bf 65}, 295 (1979). \bibitem{buni85} L. Bunimovich, %Decay of correlations in dynamical systems with chaotic %behavior {\em Sov. Phys. JETP \bf 62}, 842 (1985). \bibitem{buni95} L.A. Bunimovich, % ``Variational Principles for periodic % trajectories of hyperbolic billiards'', {\em CHAOS \bf 5}, 349 (1995). \bibitem{Biham} O. Biham and M. Kvale, % {\em Unstable periodic orbits in the stadium billiard}, {\em Phys. Rev. \bf A 46}, 6334 (1992). \bibitem{Meiss91} J.D. Meiss, % ``Regular orbits for the stadium billiard'', in {\em Quantum Chaos - Quantum Measurement}, P. Cvitanovi\'c, I. Percival, and A. Wirzba, eds. (Kluwer, Dordrecht, 1992). \bibitem{Meiss92} J.D. Meiss, % ``Cantori for the stadium billiard'', {\em CHAOS \bf 2}, 267 (1992). \bibitem{AK94} S. Akiyama and A. 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. Bensimon, M. H. Jensen and L. P. Kadanoff, {\em Phys. Rev. \bf A33}, 3622 (1986). \bibitem{pdaur2} E. Aurell, {\em Phys. Rev. \bf A34}, 5135 (1986); {\bf A35}, 4016 (1987). \bibitem{zoltan} Z. Kov\'acs, {\em J. Phys. \bf A22}, 5161 (1989). %Erik's Markovian diagrams, Pades, Feigenbaum complexes: %{\em ``Convergence of Dynamical Zeta Functions"}, \bibitem{erik} E. Aurell, to appear in {\em J. Stat. Phys.}. \bibitem{sullivan} D. Sullivan, in {\em Universality in Chaos}, 2. edition, P. Cvitanovi\'c ed., (Adam Hilger, Bristol 1989). \bibitem{18.} D. Sullivan, {\em Acta Math. \bf}, (1984) %%%%%%%%%%%%%%%%%%%%%% RENORMALIZATION FINISHED %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% FRACTALS, DIMENSIONS, %%%%%%%%%%%%%%%%%%%%%% \bibitem{100.} P. Grassberger and I. Procaccia, {\em Physica D} {\bf 13}, 34, (1984). \bibitem{101.} P. Grassberger, {\em Phys. Lett. A} {\bf 107}, 101, (1985). \bibitem{104.} G. Parisi, appendix in U. 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. Katok, {\em Liapunov exponents, entropy and periodic orbits for diffeomorphisms}, {\em Publ. Math. IHES \bf 51}, 137 (1980). \bibitem{113} D. Bessis, G. Paladin, G. Turchetti and S. Vaienti, {\em Generalized Dimensions, Entropies and Lyapunov Exponents from the Pressure Function for Strange Sets}, {\em J. Stat. Phys. \bf 51}, 109 (1988). \bibitem{D_q} P. Grassberger, {\em Phys. Lett. \bf 97A}, 227 (1983); {\bf 107A}, 101 (1985); H.G.E. Hentschel and I. Procaccia, {\em Physica \bf 8D}, 435 (1983). \bibitem{entro} P. Grassberger and I. Procaccia, {\em Phys. Rev. \bf A 31}, 1872 (1985). \bibitem{15} P. Grassberger, {\em Phys. Lett. }{\bf 97A}, 227 (1983); H. G. E. Hentschel and I. Procaccia, {\em Physica \bf 8D}, 435 (1983); R. Benzi, G. Paladin, G. Parisi and A. Vulpiani, {\em J. Phys. \bf A17}, 3521 (1984); T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia, B.I. Shraiman, {\em Phys. Rev. \bf A 33}, 1141 (1986); M.J. Feigenbaum, {\em J. Stat. 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. Series, eds., {\em Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces} (Oxford University Press, Oxford, 1991). \bibitem{Keane91} M.S. Keane, {\em Ergodic theory and subshifts of finite type}, in ref.~\cite{BKS91}. \bibitem{BowSer} R. Bowen and C. Series, {\em Publ. Math. Inst. Hautes Etud. Sci. \bf 50}, 153 (1979). % 153-170. \bibitem{Mayer92} D.H. Mayer, in G. Gy\"orgyi {\em et al.}, eds., {\em From Phase Transitions to Chaos}, pp. 521-529 (1992). \bibitem{Mayer} D.H. Mayer, {\em Lett. Math. Phys.} \bf 14\rm, 105 (1987) \bibitem{henhaus2} P. Grassberger, {\em Phys. Lett. }{\bf 97A}, 224 (1983). %recheck: is this the correl dim algorithm reference?: \bibitem{entro} P. Grassberger and I. Procaccia, {\em Phys. Rev. A} {\bf 31}, 1872 (1985). %best number for appoloninan gaskets Hausdorff dim. \bibitem{TD} Peter B Thomas and Deepak Dhar, % precise dim of Apolloninan gaskets J. Phys A 27 (1994) 2257 %%%%%%%%%%%%%%%%%%%%%% FRACTALS, DIMENSIONS FINISHED %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% YORKE-ana %%%%%%%%%%%%%%%%%%%%%% \bibitem{gre} C. Grebogi, E. Ott and J. Yorke, \psca 7D, 181, 1983 %PC this is the one with falfa including turnbacks: \bibitem{yorke2} C. Grebogi, E. Ott and J. Yorke, {\em Phys. Rev. A \bf 37, \rm 1711 (1988).} \bibitem{GOYcycles} C. Grebogi, E. Ott and J.A. Yorke, {\em Phys. Rev. \bf A36}, 3522 (1987). \bibitem{yorke} E. Ott, C. Grebogi and J.A. Yorke, {\em Phys. Lett. A \bf 135\rm, 343 (1989).} \bibitem{9.} C. Grebogi, E. Ott and J.A. Yorke, {\em Phys. Rev. \bf A36}, 3522 (1988). \bibitem{BGY} S. Bleher, C. Grebogi and E. Ott, {\em Physica \bf 46D}, 87 (1990). % extensive chaotic scattering paper %Chaotic scattering \bibitem{lg}Y.-Ch. Lai, C. Grebogi, Phys.Rev. E {\bf 49}, 3761 (1994); Y.-Ch. Lai {\em et al.}, Phys.Rev.Lett. {\bf 71}, 2212 (1993) \bibitem{BGY} Kan, Kocak and J.A.~Yorke, %"Antimonotonicity: Concurrent Creation and Annihilation of Periodic Orbits" to appear in Annals of Math. (July 91) %transient chaos review \bibitem{Telrev} T. T{\'e}l, %"On the organization of transient chaos: Application to irregular scattering" {\em hys. \bf ??}, ?? (1989). %%%%%%%%%%%%%%%%%%%%%% Yorke-ana FINISHED %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% INTERMITENCY %%%%%%%%%%%%%%%%%%%%%% \bibitem{prell}Thomas Prellberg, "Maps of Intervals with Indifferent Fixed Points: Thermodynamic Formalism and Phase Transitions", dissertation, Virginia Polytechnic Institute and State University (June 1991). % now Australia, Dept of Math Univ of %Melbourne.) It contains some very simple but efficient ideas to %overcome "marginal instabilities" in some cases. (Essentially, %look at a first return map which is expanding, and compare the %transfer for the original map with the renormalized transfer.) %thermodynamic formalism and phase transitions for maps %with marginal fixed points: \bibitem{PS92} T. Prellberg and J. Slawny, %``Maps of Intervals with Indifferent Fixed-Points - %Thermodynamic Formalism and Phase-Transitions {\em J. Stat. Phys. \bf 66}, 503 %-514 (1992). \bibitem{tak81} Y. Takahashi, %``Fredholm determinant of unimodal map'' {\em Science Papers of Coll. Ed. Univ. Tokyo \bf 8}, 61 %-87 (1981). \bibitem{artuso2} R. Artuso, {\em J. Phys. \bf A21}, L923 (1988). %related to J Phys A Letter; name for the Riemman-like function \bibitem{FK} B. Fornberg and K.S. K\"olbig, {\em Math. of Computation \bf 29\rm, 582 (1975)} %The reflection formula for the Lerch trancendent is given in %eq.(9.552) on p.1075 in Gradshteyn and Ryzhik. \bibitem{INT} X.-J. Wang, {\em Phys. Rev. }{\bf A 39}, 3214 (1989); {\bf A 40}, 6647 (1989) % which look very much like Artuso's little paper % on Fisher like phase transition \bibitem{Yuri} Michiko Yuri, %``Invariant measures for certain multi-dimeansional maps'' {\em Nonlinearity}, (1984). S.Grossman og H.Horner, % ``Long time correlations in discrete Chaotic Dynamics" \bibitem{hhrugh92} H.H. Rugh, {\em ``The correlation spectrum for hyperbolic analytic maps''}, {\em Nonlinearity \bf 5}, 1237 (1992). % 1237--1263 \bibitem{hhrugh92} H.H. Rugh, {\em ``Fredholm determinants for real-analytic hyperbolic diffeomorphisms of surfaces''}, p. 297, %rm 297--303 in {\em XIth International Congress of Mathematical Physics (Paris, 1994)} (Internat. Press, Cambridge, 1995). \bibitem{hhrugh96a} H.H. Rugh, {\em ``Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems''}, {\em Ergodic Theory Dynamical Systems \bf 16}, 805 (1996). % 805--819 \bibitem{hhrugh96b} H.H. Rugh, {\em ``Intermittency and regularized Fredholm determinants''}, {\tt chao-dyn/9610011}. %next bunch from S. Isola http://mpej.unige.ch/mp_arc/papers/00-466 % On systems with finite ergodic degree, Nonlinearity (2002) % \bibitem[Aa]{Aa} {\sc J Aaronson}, {\it An introduction to infinite ergodic theory}, AMS, 1997. \bibitem[Ab]{Ab} {\sc L M Abramov}, \, {\sl The entropy of a derived automorphism}, Amer. Math. Soc. Transl. (2) {\bf 49} (1965), 162-166. \bibitem[Ba1]{Ba1} {\sc V Baladi}, \, {\sl Positive Transfer Operators and Decay of Correlations}, World Scientific, 2000. \bibitem[Ba2]{Ba2} {\sc V Baladi}, \, {\sl Dynamical zeta functions}, Real and Complex Dynamical Systems (B. Branner and P. Hjorth eds.), Kluwer Academic Publishers, 1995. \bibitem[Bo]{Bo} {\sc R Bowen}, \, {\it Equilibrium states and the ergodic theory of Anosov diffeomorphisms}, LNM 470 (1975), Springer-Verlag. \bibitem[CI]{CI} {\sc M Campanino, S Isola}, \, {\sl Statistical properties of long return times in type I intermittency}, Forum Mathem. {\bf 7} (1995), 331-348. \bibitem[CI2]{CI2} {\sc M Campanino, S Isola}, \, {\sl Infinite invariant measures or non-uniformly expanding transformations of $\ui$: weak law of large numbers with anomalous scaling}, Forum Mathem. {\bf 8} (1996), 71-92. \bibitem[Che]{Che} {\sc N Chernov}, {\sl Limit theorems and markov approximations for chaotic dynamical systems}, Probab. Theory Relat. Fields {\bf 101}, (1995) 321-362. \bibitem[Chu]{Chu} {\sc K L Chung}, {\it Markov chains with stationary transition probabilities}, Springer 1967. \bibitem[Fe]{Fe} {\sc W Feller}, \, {\sl An Introduction to Probability Theory and Its Applications}, Volume 2, J.Wiley and Sons, New York 1970. \bibitem[FF]{FF} {\sc B U Federhof, M E Fisher}, Annals of Physics (N.Y.) {\bf 58} (1970). \bibitem[FL]{FL} {\sc A M Fisher, A Lopes}, {\sl Polynomial decay of correlation and the central limit theorem for the equilibrium state of a non-H\"older potential}, Preprint. \bibitem[Ga]{Ga} {\sc G Gallavotti}, \, {\sl Funzioni zeta e insiemi basilari}, Accad. Lincei Rend. Sc. fis. mat. e nat. {\bf 61} (1976), 309-317. \bibitem[Har]{Har} {\sc G H Hardy}, {\it Divergent series}, Oxford at the Calrendon Press 1949. \bibitem[Hay]{Hay} {\sc N T A Haydn}, {\sl Meromorphic extension of the zeta function for Axiom A flows}, Erg. Th. Dyn. Sys. {\bf 10} (1990), 347-360. \bibitem[HI]{HI} {\sc N T A Haydn, S Isola}, {\sl Parabolic rational maps}, to appear in J. London Math. Soc. \bibitem[Ho]{Ho} {\sc F Hofbauer}, {\sl Examples for the nonuniqueness of the equilibrium state}, Trans. Amer. Math. Soc. {\bf 228} (1977), 223-241. \bibitem[Is1]{Is1} {\sc S Isola}, {\sl Renewal sequences and intermittency}, J. Stat. Phys. {\bf 97} (1999), 263-280. \bibitem[Is2]{Is2} {\sc S Isola}, {\sl On the rate of convergence to equilibrium for countable ergodic Markov chains}, 1999 Preprint. \bibitem[Ka]{Ka} {\sc T Kato}, {\it Perturbation theory of linear operators}, Springer-Verlag, Berlin Heidelberg New York (1980). \bibitem[Ke]{Ke} {\sc G Keller}, {\sl On the rate of convergence to equilibrium in one-dimensional systems}, Comm. Math. Phys. {\bf 96} (1984), 181-193. \bibitem[Ki]{Ki} {\sc J F C Kingman}, {\it Regenerative Phenomena}, John Wiley, 1972. \bibitem[LSV]{LSV} {\sc C Liverani, B Saussol, S Vaienti}, {\sl A probabilistic approach to intermittency}, Erg. Th. Dyn. Sys. {\bf 19} (1999), 671-685. \bibitem[MRTVV]{MRTVV} {\sc C Maes, F Redig, F Takens, A Van Moffaert, E Verbitsky}, {\sl Intermittency and weak Gibbs states}, Nonlinearity {\bf 13} (2000), 1681-1698. \bibitem[Nu]{Nu} {\sc R Nussbaum}, {\sl The radius of the essential spectrum}, Duke Math. J. {\bf 37} (1970), 473-478. \bibitem[Pol1]{Pol1} {\sc M Pollicott}, {\sl Meromorphic extensions of generalised zeta functions}, Invent. math. {\bf 85} (1986), 147-164. \bibitem[Pol2]{Pol2} {\sc M Pollicott}, {\sl Rates of mixing for potentials of summable variation}, Preprint 1998. \bibitem[PP]{PP} {\sc W Parry, M Pollicott}, \, {\it Zeta functions and the periodic orbit structure of hyperbolic dynamics}, Soci\'et\'e Math\'ematique de France (Ast\'erisque {\bf 187-188}), Paris. \bibitem[Pos]{Pos} {\sc A G Postnikov}, \, {\it Tauberian Theory and its Applications}, Proceedings of the Steklov Institute of Mathematics, 1980, Issue 2. \bibitem[PS]{PS} {\sc T Prellberg, J Slawny}, {\sl Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions}, J. Stat. Phys. {\bf 66} (1992), 503-514. \bibitem[RS]{RS} {\sc M Reed, B Simon}, \,{\it Methods of Modern Mathematical Physics}, Vol. IV: Analysis of Operators, Academic Press, New York 1978. \bibitem[Ro]{Ro} {\sc B A Rogozin}, {\sl An estimate of the remainder term in limit theorems of renewal theory}, Theory Prob. Appl. {\bf 18} (1973), 662-677. \bibitem{BowRue} R. Bowen and D. Ruelle, {\em Inventions. Math. } {\bf ??}, ?? (1975). % Ergodic theory of Axiom A flows. Explains % how to construct invariant measures from % transfer op. Case of unstable jacobian % weight explained for flows. (See also % Ledrappier-Young and Ruelle Amer J. Math 1976.) \bibitem{113} F. Ledrappier and D. Ruelle, Amer J. Math (1976). % Grothendieck theory for expanding maps % AND Anosov flows applied to some zeta % functions via Fredholm determinants. % No measure theory/ correlation functions. \bibitem{Ruelle76} D. Ruelle, {\em ``Zeta functions for epanding maps and Anosov flows}, {\em Inventiones math. \bf 34\rm, 231 (1976)}. \bibitem{21 }same % ``Locating resonances for Axiom A dynamical systems" {\em J. Stat. Phys. \bf 44 }, 281--292 (1986) \bibitem{22 }same % ``The thermodynamic formalism for expanding maps" {\em Comm. Math. Phys. \bf 125 }, 239--262 (1989) \bibitem{BR94} V. Baladi and D. Ruelle, {\em ``An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps''}, {\em Ergodic Theory Dynamical Systems \bf 14}, 621 %-632 (1994). \bibitem{R95} D. Ruelle, {\em ``Functional equation for dynamical zeta functions of Milnor-Thurston type''}, {\em Comm. Math. Phys. \bf 175}, 63 (1996). % \pages 63--88 \bibitem{R95a} D. Ruelle, {\em ``Sharp determinants for smooth interval maps''}, Proceedings of Montevideo Conference 1995, IHES preprint (March 1995). \bibitem{BR96} V. Baladi and D. Ruelle, {\em ``Sharp determinants''}, {\em Invent. Math. \bf 123}, 553 %-574 (1996). \bibitem{BKRS96} V. Baladi, A. Kitaev, D. Ruelle, and S. Semmes, {\em ``Sharp determinants and kneading operators for holomorphic maps''}, IHES preprint (1995). \bibitem[Ru1]{Ru1} {\sc D Ruelle}, \,{\sl Zeta functions for expanding maps and Anosov flows}, Invent. Math. {\bf 34} (1976), 231-242. \bibitem[Ru2]{Ru2} {\sc D Ruelle}, {\sl Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval}, American Mathematical Society (CRM Monograph Series, {\bf 4}), Providence, Rhode Island USA, 1994. \bibitem[Ru3]{Ru3} {\sc D Ruelle}, \,{\sl One dimensional Gibbs' states and Axiom A diffeomorphisms}, J. Diff. Geom. {\bf 25} (1987), 117-137. \bibitem[Ru4]{Ru4} {\sc D Ruelle}, \,{\it Thermodynamic Formalism}, Addison-Wesley Publ. Co. 1978. \bibitem[Rug]{Rug} {\sc H H Rugh}, {\sl Intermittency and Regularized Fredholm Determinants}, Invent. Math. {\bf 135} (1999), 1-24. \bibitem[Si]{Si} {\sc B Simon}, \,{\it The Statistical Mechanics of Lattice Gases}, Princeton University Press, 1993. \bibitem[Th]{Th} {\sc M Thaler }, {\it Estimates of the invariant densities of endomorphisms with indifferent fixed points}, Israel Jour. Math. {\bf 37} (1980), 303-314. \bibitem[Wal1]{Wal1} {\sc P Walters}, {\sl Ruelle's operator theorem and $g$-measures}, Trans. Amer. Math. Soc. {\bf 214} (1975), 375-387. \bibitem[Wal2]{Wal2} {\sc P Walters}, {\sl Invariant measures and equilibirum states for some mappings which expand distances}, Trans. Amer. Math. Soc. {\bf 236} (1978), 121-153. \bibitem[Yo]{Yo} {\sc L S Young}, {\sl Recurrence times and rate of mixing}, Isr. J. Math. {\bf 110} (1999), 153-188. \bibitem[Yu]{Yu} {\sc M Yuri}, {\sl Thermodynamic formalism for certain non-hyperbolic maps}, Erg. Th. Dyn. Sys. {\bf 19} (1999), 1365-1378. \bibitem[Zig]{Zig} {\sc A Zigmund}, \,{\it Trigonometric Series}, Cambridge at the University Press, 1968. %%%%%%%%%%%%%%%%%%%%%% INTERMITENCY FINISHED %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% SYMBOLIC DYNAMICS %%%%%%%%%%%%%%%%%%%%%% \bibitem{78.} For a very readable survey of complexity and symbol sequences, see the introduction to: M. Nordahl, {\em Thesis}, Chalmers Institute of Technology, G\"oteborg, Sweden (1988). \bibitem{107.} V.M. Alekseev and M.V. Jakobson, {\em Symbolic Dynamics and Hyperbolic Dynamical Systems}, Physics Reports, {\bf 75}, 287, (1981). \bibitem{dynzet} See ref. \cite{ruelle}, sect. 7.23. \bibitem{erikz} E. Aurell, %Erik's Markovian diagrams, Pades, Feigenbaum complexes: %{\em ``Convergence of Dynamical Zeta Functions"}, {\em J. Stat. Phys. \bf 58}, 967 (1990). \bibitem{symp} E. Aurell, {\em G\"oteborg preprint 89--10}, submitted to {\em Phys. Rev. A}. \bibitem{KTH92a} K.T. Hansen, % ``Remarks on the symbolic dynamics for the H\'enon map" {\em Phys. Lett. \bf A 165}, 100 (1992) \bibitem{#} B. Eckhardt, and D. Wintgen, %``Indexes in Classical Mechanics {\em J. Phys. \bf A 24}, 4335 (1991) \bibitem{troll} G. Troll % A devil's staircase into chaotic scattering. {\em Pysica D \bf 50}, 276 (1991) \bibitem{riordan} J. Riordan, {\em An Introduction to Combinatorial Analysis} ( Wiley, New York 1958). ; E.N. Gilbert and J. Riordan, {\em Illinois J.Math} {\bf 5}, 657 (1961). \bibitem{brucks} K.M. Brucks, {\em Adv. Appl. Math.} {\bf 8}, 434 (1987). %this has some automata (not all): \bibitem{gras86} P. Grassberger, %``Toward a quantitative theory of % self-generated Complexity {\em Int. J. Theor. Phys \bf 25}, 907 (1986). \bibitem{IP} S. Isola and A. Politi, % Universla encoding for unimodal maps (markov diagrams ala Kai) {\em J. Stat. Phys. } {\bf 61}, 259 (1990). \bibitem{AP} G. D'Alessandro and A. Politi, % Hierarchical approach to Complexity ... {\em Phys. Rev. Lett. \bf 64}, 1609 (1990). \bibitem{WX94} Y. Wang and Huimin Xie, % ``Grammatical complexity of unimodal maps with eventually periodic kneading sequences'', {\em Nonlinearity } {\bf 7}, 1419 (1994). %%%%%%%%%%%%%%%%%%%%%% SYMBOLIC DYNAMICS FINISHED %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% SCATTERING %%%%%%%%%%%%%%%%%%%%%%%%%% \bibitem{KT} L. Kadanoff and C. Tang, % {\em Escape from strange repellers}, {\em Proc. Natl. Acad. Sci. USA, \bf 81}, 1276 (1984). %transient chaos - irregular scattering falfaing \bibitem{Tel_scat} T. T{\'e}l, {\em J. Phys. \bf A22}, L691 (1989). \bibitem{Tel_rev} T. T{\'e}l, %transient chaos review %"On the organization of transient chaos: %Application to irregular scattering" in Bai-lin Hao, ed., {\em Directions in Chaos}, vol. 3, (World Scientific, Singapore 1988) p. 149. %--221 \bibitem{CHAOS93} T. T{\'e}l and E. Ott, eds., {\em Chaotic Scattering - theme issue}, {\em CHAOS \bf 2}, 417-782 (1993). \bibitem{tdisk} B. Eckhardt, {\em Fractal properties of scattering singularities}, {\em J. Phys. \bf A 20\rm, 5971 (1987).} \bibitem{IS_rev1} B. Eckhardt, {\em Physica \bf D 33}, 89 (1988). \bibitem{SMYO} Shigematsu H, Mori H, Yoshida T and Okamoto H 1983 {\em J. Stat. Phys.} {\bf 30} 649%-- \bibitem{EGP} Eckhardt B, Gomez JM and Pollak E 1990 {\em Chem. Phys. Lett.} {\bf 174} 325%--332 \bibitem{dorfle} D\"orfle M 1985 {\em J. Stat. Phys.} {\bf 40} 93%--132 \bibitem{Morita} T. Morita, {\em Trans. Am. Math. Soc., \bf 325\rm, 819 (1991)} \bibitem{ER92} B. Eckhardt and G. Russberg, {\em Resummations of classical and semiclassical periodic orbit expressions}, {\em Phys. Rev. \bf E 47}, 1578 (1993). % poles in Selberg products found here \bibitem{faulkner} J.S. Faulkner, {\em Scattering theory and cluster calculations}, J.Phys. {\bf C 10 } (1977) 4661-4670. \bibitem{moroz} A. Moroz, {\em Density-of-states calculations and multiple-scattering theory for photons}, Phys. Rev. {\bf B51} (1995) 2068. Time-delay J. M. Jauch, K. B. Sinha and B.N. Misra, {\em Time-Delay in Scattering Processes} Helv. Phys. Acta {\bf 45} (1972) 398-426. Ph. Martin and B. Misra, ``On trace-class operators of scattering theory and the asymptotic behavior of scattering cross section at high energy'', J. Math. Phys. 14 (1973) 997-1005. E. Seiler, Comm. Math. Phys. 42 (1975) 163-182 E. Seiler and B. Simon, J. Math. Phys. 16 (1975) 2284-93. \bibitem{Lloyd} P. Lloyd, {\em Wave propagation through an assembly of spheres. II. The density of single-particle eigenstates}, Proc. Phys. Soc. {\bf 90} (1967) 207-216. \bibitem{Lloyd_smith} P. Lloyd and P.V. Smith, {\em Multiple-scattering theory in condensed materials}, Adv. Phys. {\bf 21} (1972) 69-142 and references therein. %%%%%%%%%%%%%%%%%%%%%% SCATTERING FINISHED %%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% EXTERNAL NOISE: \bibitem{CH}J.P. Crutchfield and B.A. Huberman, {\em Phys. Lett. \bf 77A}, 407 (1980), reprinted in ref. \cite{u_in_c}. \bibitem{SWM} B. Shraiman, C. E. Wayne and P. C. Martin, \PRL{46}, 935 (1981), reprinted in ref. \cite{u_in_c}. \bibitem{CNR} J. Crutchfield, M.Nauenberg and J. Rudnick, \PRL{46}, 933 (1981), reprinted in ref. \cite{u_in_c}. \bibitem{FH} M.J. Feigenbaum and B. Hasslacher, {\em Phys. Rev. Lett. \bf 49}, 605 (1982). % noise problem treated in terms of path integrals. \bibitem{Watanabe87} S. Watanabe, %{\em ``Analysis of Wiener Functionals (Malliavin % calculus) and its applications to heat kernels} {\em Ann. of Prob. \bf 15} (1987) 1. % 1-39 is the one that introduced expansions in fat delta. It also talks about stochastic flows in terms of Riemann metric, connections etc, like w3fusion.ph.utexas.edu/~jeanluc/ when formulating the Langevin continuous flow in several dimensions. %%%%%%%%%%%%%%%%%%%%%% EXTERNAL NOISE FINISHED %%%%%%%%%%%%%% %%%%%%%%%%%%%% DIFFUSION CONFUSSION %%%%%%%%%%%%%%%%%%% \bibitem{art91} R. Artuso ``Diffusive dynamics and periodic orbits of dynamical systems", {\em Phys. Lett. \bf A 160}, 528 (1991). \bibitem{ACL93} R. Artuso, G. Casati and R. Lombardi, ``Periodic orbit theory of anomalous diffusion'', {\em Phys. Rev. Lett. \bf 71}, 62 (1993). \bibitem{ACL94} R. Artuso, G. Casati and R. Lombardi, % just conference proceedings {\em Physica \bf A 205}, 412 (1994). \bibitem{A94} R. Artuso, % just conference proceedings {\em Physica \bf D 76}, 101 (1994). \bibitem{vance} W.N. Vance, ``Unstable periodic orbits and transport properties of nonequilibrium steady states'', {\em Phys. Rev. Lett. \bf 96}, 1356 (1992). \bibitem{GT77} S. Grossmann and S. Thomae, {\em Z. Naturforsch. \bf 32 a}, 1353 (1977) %ARTUSO%VAXMI@nbivax.nbi.dk Mon May 6 17:31 MET 1991 % Diffusive Dynamics and Periodic Orbits of Dynamical Systems \ref \no 8 \by R.W. Leven and B.P. Koch \jour Phys.Lett. \vol 86A \pages 71 (1981 ) \ref \no 9 \by B.V. Chirikov \jour Phys.Rep. \vol 52 \pages 263 (1979 ) \bibitem{J} B.A. Huberman, J.P. Crutchfield and N.H. Packard, {\em Appl.Phys.Lett.} {\bf 37} (1980) 750 M. Cirillo and N.F. Pedersen, {\em Phys.Lett.} {\bf 90A} (1982) 150 E. Ben-Jacob, J. Goldhirsch, Y. Imry and S. Fishman, {\em Phys.Rev.Lett.} {\bf 49} (1982) 1599 \bibitem{CE} R. Mainieri, %``Thermodynamic-Zeta Functions for Ising-Models with Long- % Range Interactions {\em Phys.Rev.} {\bf A45} (1992) 3580 \bibitem{BSM} B. Eckhardt, % {\em Periodic Orbits and Diffusion in Standard Maps}, % {\em Marburg preprint} (July 1992) {\em Phys. Lett. \bf 172A} 411 (1993). \bibitem{MP} Y. Pomeau and P. Manneville, {\em Commun. Math. Phys.} {\bf 74} (1980) 189; P. Manneville, {\em J. Phys.} (Paris) {\bf 41} (1980) 1235 \bibitem{UH} B. Friedman and R.F. Martin, Jr., {\em Phys. Lett.} {\bf 105A} (1984) 23 \bibitem{Bleher} P.M. Bleher, % ``Statistical properties of two-dimensional Lorentz gas % with infinite horizon" {\em J.Stat.Phys.} {\bf 66} (1992) 315 %end of ARTUSO Sep 23 1992 referencess \bibitem{lore} H.A. Lorentz, %Lorentz gas introduced here {\em Proc. Amst. Acad. \bf 7}, 438 (1905). \bibitem{MZ} J. Mechta and R. Zwanzig, %explicit numbers computed from simulations here %"Diffusion in a periodic Lorentz gas" {\em Phys. Rev. Lett. \bf 50}, 1959 (1983). \bibitem{mechta} J. Mechta %"Power law decay of Correlations in a Billiard problem" {\em J. Stat. Phys. \bf 33}, 555 (1983). \bibitem{sina70} Ya.G. Sinai, %Sinai billiards introduced here {\em Usp. Mat. Nauk \bf 25}, 141 (1970). \bibitem{BS80} L. Bunimovich and Ya.G. Sinai, % shows that decays are funny exponentials %{\em Markov Partition for Dispersed Billiard}, {\em Comm. Math. Phys. \bf 78}, 247 (1980); {\bf 78}, 479 (1980); {\em Erratum, ibid. \bf 107}, 357 (1986). \bibitem{GG93} Garrido Pedro, Gallavotti Giovanni %``Billiards correlation functions'' {\em J. Stat. Phys. \bf ??}, 549 (1984). %Paper: chao-dyn/9310005 27 Oct 93 %From: giovanni@boltzmann.rutgers.edu (Giovanni Gallavotti) % experiments on the time decay of velocity % autocorrelation functions in billiards % results which are compatible with an exponential mixing hypothesis, % first put forward by [FM]: they do not seem compatible with the % stretched exponentials believed, in spite of [FM], to describe the % mixing. D. Weiss et al. "Electron Pinball ...", PRL 66, 2790 (1991). (they can build little sinai lattices) % 16 Dec 91 : Peter Koch, Stony Brook, possible diffusion experiments: S. Stridhar, "Experimental obs. on scars" PRL 67, 785 (1991). E. Yablonovitch, "Photonic band structure...", PRL 63, 1950 (1989). S. John, "The Localization of Light...", Comments Cond. Mat. Phys 14, 193 (1988). I.S. Graham et al, "Experiments ... Acoustic...", PRL 64, 3135 (1990). H. Muller, exps. on light propagation in disordered media, probably in Comments Atomic Mol. Optical Phys. 1990 or 1991 M.L. Roukes, A. Scherer and B.P. Van der Gaag, "Are Transport Anomalies in ``Electron Waveguides" Classical?" PRL 64, 1154 (1990). M.L. Roukes and O.L. Alerhand, "Mesoscopic Junctions, Random Scattering and Strange Repellers" PRL 65, 1651 (1990). \bibitem{BH91} C.W.J. Beenakker and V. van Houten, {\em Quantum transport in semiconductor nanostructures}, in \cite{ET}. \bibitem{ET} H. Echenreich and D. Turnbull, eds., {\em Solid state physics - semiconductor heterostructures and nanostructures}, % 17 Dec 91: Our eternal ``itermittency", ``power laws" , ...., problems %are in this context called diffusion anomalies, and might have something %to do with Levy flights and similar. Perhaps E.W. Montroll and M.E. Shlesinger, ``On the Wonderful World of Random Walks", in J.L. Lebowitz and E.W. Montroll, eds., {\sl Nonequilibrium Phenomena II; from Stochastics to Hydrodynamics} (North-Holland, Amsterdam, 1984). %offers some ideas how to look at anomalous diffusion.... \bibitem{JBS} R.A. Jalabert, H.U. Baranger and A.D. Stone, %conductance fluctuations in the balistic regime - %probe for quantum chaos? % ballistic conductors reviewed in \bibitem{RK} M.A. Reed and W.P. Kirk, eds., {\em Nanostructure Physics and Fabrication} (Academic Press, New York, 1989). submitted to {\em Phys. Rev. Lett. } (June 1990). \bibitem{Piq90} J.P. Pique, {\em J. Opt. Soc. Am.} {\bf B 7}, 1819 (1990) %% next bunch is from \bibitem{KD} R. Klages, J.R. Dorfman %% Simple Maps with Fractal Diffusion Coefficients %% e-mail: rkla0433@w421zrz.physik.tu-berlin.de % 1-d maps: \bibitem{gro} S. Grossmann, H. Fujisaka, ``Diffusion in discrete nonlinear dynamical systems'', Phys.Rev. A {\bf 26}, 1179 (1982); H. Fujisaka, S. Grossmann, Z.Phys. B {\bf 48}, 261 (1982) \bibitem{sfk}M. Schell, S. Fraser, R. Kapral, `` Diffusive dynamics in systems with translational symmetry: a one--dimensional--map model'' Phys.Rev. A {\bf 26}, 504 (1982) % they ``predict'' Diffusion constant, OK close to treshold \bibitem{gg}T. Geisel, J. Nierwetberg, ``Onset of diffusion and universal scaling in chaotic systems'' Phys.Rev.Lett. {\bf 48}, 7 (1982); S. Grossmann, S. Thomae, Phys.Lett. {\bf 97A}, 263 (1983); \bibitem{10} T. Geisel and J. Nierwetberg, Phys. Rev. Lett. 47, 975 (1981). \bibitem{GT} T. Geisel and S. Thomae, {\em Phys.Rev.Lett.} {\bf 52}, 1936 (1984). \bibitem{GNZ} T. Geisel, J. Nierwetberg and A. Zacherl, {\em Phys.Rev.Lett.} {\bf 54}, 616 (1985). A. Zacherl, T. Geisel, J. Nierwetberg and G. Radons, {\em Phys.Lett.} {\bf 114A}, 317 (1986). V. Urumov and L. Kocarev, Phys. Lett. A 144, 220 (1990). %2-d standard map numerical diffusion: \bibitem{rw}A.B. Rechester, R.B. White, Phys.Rev.Lett. {\bf 44}, 1586 (1980); A.B. Rechester, M.N. Rosenbluth, R.B. White, Phys. Rev. A {\bf 23}, 2664 (1981) \bibitem{cm}J.R. Cary, J.D. Meiss, A. Bhattacharjee, Phys.Rev. A {\bf 23}, 2744 (1981); J.R. Cary, J.D. Meiss, Phys. Rev. A {\bf 24}, 2664 (1981); T.M. Antonsen and E. Ott, Phys.Fluids {\bf 24}, 1635 (1981) \bibitem{MKMP} R.S.~MacKay, J.D. Meiss and I. Percival, {\em Physica D} {\bf 13}, 55 (1984). \bibitem{DMP} I. Dana, N.W. Murray and I. Percival {\em Phys. Rev. Lett. \bf 62}, 233 (1989). \bibitem{dana89} I. Dana, % Hamiltonian transport on unstable periodic orbits {\em Physica \bf D 39}, 205 (1989) \bibitem{dana} I. Dana, % Organization o fchaos in area-preserving maps {\em Phys. Rev. Lett. \bf 64}, 2339 (1990). % Entropy+Lyapunovs --> transport coefficients for thermostats \bibitem{BEC} A. Baranyai, D.J. Evans and E.G.D. Cohen, ``Field-Dependent Conductivity and Diffusion in a Two-Dimensional Lorentz Gas'' {\em J. Stat. Phys. \bf 70}, 1085 (1993). \bibitem{ech}D.J. Evans, E.G.D. Cohen, G.P. Morris, Phys. Rev. A {\bf 42}, 5990 (1990); N.I. Chernov {\em et al.}, Phys. Rev. Lett. {\bf 70}, 2209 (1993); Comm. Math. Phys. {\em 154}, 569 (1993); H.A. Posch, W.G. Hoover, Phys. Lett. A {\bf 123}, 227 (1987); Phys. Rev. A {\em 39}, 2175 (1989) %construction of finite Markov partitions: \bibitem{boy}see, e.g., A. Boyarski, M. Skarowsky, Trans. Am. Math. Soc. {\bf 225}, 243 (1979); A. Boyarski, J.Stat. Phys. {\bf 50}, 213 (1988); %discussion of finite Markov partitions: \bibitem{bst}C.S. Hsu, M.C. Kim, Phys. Rev. A {\bf 31}, 3253 (1985); N. Balmforth, E.A. Spiegel, C. Tresser, Phys.Rev.Lett. {\bf 72}, 80 (1994) %bloc-circulant matrices: \bibitem{bk}T.H. Berlin, M. Kac, Phys.Rev. {\bf 86}, 8211 (1952); see also \cite{PJD} %transport by turnstiles: \bibitem{mch} R.S. Mackay, J.D. Meiss, I.C. Percival, Physica D {\bf 13}, 55 (1984), Q. Chen, J.D. Meiss, Nonlinearity {\bf 39}, 347 (1989); Q. Chen {\em et al.}, Physica D {\bf 46}, 217 (1990); J.D. Meiss, %``Symplectic Maps, Variational-Principles, and Transport Rev.Mod.Phys. {\bf 64}, 795 (1992) \bibitem{exac_diff} H.-C. Tseng, H.-J. Chen, P.-C. Li, W.-Y. Lai, C.-H. Chou and H.-W. Chen, ``Some exact results for the diffusion coefficients of maps with pruned cycles'', {\em Phys. Lett. \bf A 195}, 74 (1994). %see my referee report 21/7-94 for imprved version %next paper seems very simlar \bibitem{CCC} C.-C. Chen, %Chia-Chu Chen %National Chung-Hsing University, Taiwan % chiachu@phys2.nchu.edu.tw ``Diffusion Coefficient of Piecewise Linear Maps'', % National Chung-Hsing Univ. preprint (March 1994) {\em Phys. Rev. \bf E 51}, 2815 (1995). \bibitem{MR94} G.P. Morriss and L. Rondoni, {\em J. Stat. Phys. \bf 75}, 553 (1994). % Lorentz Gas \bibitem{LRM94} J. Lloyd, L. Rondoni and G.P. Morriss, ``The Breakdown of Ergodic Behaviour in the Lorentz Gas'', (submitted). \bibitem{RMLNC94} L. Rondoni, G.P. Morriss, J.P. Lloyd, M. Niemeyer, and E.G.D. Cohen, ``Lorentz Gas, Periodic Orbit Expansions, Partitions, and Ergodicity'', {\em Chaos, Solitons \& Fractals}, (in press). \bibitem{MRC94} G.P. Morriss, L. Rondoni and E.G.D. Cohen, ``A Dynamical Partition Function for the Lorentz Gas'', (submitted). \bibitem{LNRM94} J. Lloyd, M. Niemeyer, L. Rondoni and G.P. Morriss, ``The Nonequilibrium Lorentz Gas'', Univ. of New South Wales preprint (Sept. 1994). \bibitem{DM97}C.P. Dettmann and G. P. Morriss, Phys. Rev. Lett. {\bf 78}, 4201 (1997). \bibitem{HBA94} A. Hakmi, F. Bosco and I. Antoniou, ``The First Return Map of the Periodic Lorentz Gas'', ULB, Bruxelles prperint (aug. 1994). \bibitem{Liverani} C. Liverani, ``Decay of correlations for piecewise expanding maps'', U. of Rome prperint (aug. 1994). H. Haken and G. Mayer-Kress, ``Chapman-Kolmogorov Eq. and .. '' {\em Z. f. Physik \bf B 43}, 185 (1981). % study of iterated mappings with additive/multiplicative % noise. Write Chapman-Kolmogorov equation (say that the Fokker-Planck % is inadequate) which is what we would call ``Fokker-Planck'', % say that is is of form of a Fredholm integral equation. % Iterated, they call it path integral similar to the one % introduced in \cite{SWM}. Very formal, we do not need it. M. Roncadelli, ``Small-fluctuation expansion ...'', {\em Phys. Rev. \bf E 52}, 4661 (1995). % rewrites Fokker-Planck in terms of Wiener-Onsager-Mahlup % Lagrangian; constructs weak diffusion coefficient % saddle-point expansion; writes transport equations for % higher order coefficients. Might be a useful reference G. Ryskin, {\em Phys. Rev. \bf E 56}, 5123 (1997). % says Kramers-Moyal expansion, Fokker-Planck equation not right % proposes alternative equation which is Galilean invariant. % Looks pretty formal. \bibitem{Onsager53} L.~Onsager and S.~Machlup, % ``Fluctuations and Irreversible Processes", {\em Phys. Rev. \bf 91}, 1505, 1512 (1953). %%%%%%%%%%%%%% DIFFUSION CONFUSSION FINISHED %%%%%%%%%%%%%% %%%%%%%%%%%%%% MESOSCOPICS %%%%%%%%%%%%%%%%%%%%%%% \bibitem{HvO94} G. Hackenbroich and F. von Oppen, ``Semiclassical theory of transport in antidot lattices'', {\em Z. Phys. \bf B 97}, 157 (1995). %%%%%%%%%%%%%% MESOSCOPICS FINISHED %%%%%%%%%%%%%% \bibitem{katzen} D. Katzen and I. Procaccia, {\em Phys. Rev. Lett} {\bf 58}, 1169 (1987). \bibitem{BR87} T. Bohr and D. Rand, {\em``The entropy function for characteristic exponents}, {\em Physica \bf 25D}, 387 (1987). \bibitem{6.} P. Sz\'epfalusy, T. T\'el, A. Csord\'as and Z. Kov\'acs, {\em Phys. Rev. \bf A 36}, 3525 (1987). \bibitem{ozorio} G.L. Da Silva Ritter, A.M. Ozorio de Almeida and R. Douady, {\em Physica \bf D29\rm, 181 (1987).} %the ``sum rule'' for probability conservation is here \bibitem{HOdA84} J.~H.~Hannay and A.~M.~Ozorio de Almeida, {\em J. Phys. \bf A 17}, 3429, (1984). \bibitem{ruelcont} A lucid introduction to escape from repellers is given in L.P. Kadanoff and C. Tang, {\em Proc. Natl. Acad. Sci. \bf 81\rm, 1276 (1984)}. \bibitem{4.} R. Badii and A.Politi, {\em Physica Scripta \bf 35}, 243 (1987). \item{36.}P. Grassberger, {\riv J.Stat.Phys. \aint 26}, 173 (1981). \bibitem{BR56} J. Balatoni and A. Renyi, {\em Publi. Math. Inst. Hung. Acad. Sci. \bf 1}, 9 (1956); (english translation in , Vol. 1, p. 588 (Akademia Budapest, 1976)); A. Renyi, (appendix) (North- Holland, 1970). %%%%%%%%%%%%%%%%%%%%% TURBULENCE %%%%%%%%%%%%%%%%%%%%%%%%%%% 18. B. B. Mandelbrot. J. Fluid Mech. 62, 331 (1974). %3 papers on laminar boundary layer flow: \bibitem{Kleb} P.S. Klebanoff, et al. {\em J. Fluid Mech. \bf 12}, 1 (1962). \bibitem{kach94} Y.S. Kachanov, et al. {\em Fluid Dyn. \bf 12}, 283 (1978). \bibitem{kach94} Y.S. Kachanov, %Theo. and Applied Mechanics, Novosibirsk %``Physical mechanisms of laminar-turbulent transition'', {\em Ann. Rev. Fluid Mech. \bf 26}, ?? (1994). \bibitem{KawKida01} G. Kawahara and S. Kida, ``Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst,'' {\it J. Fluid Mech.} {\bf449}, 291 (2001). % pp. 291-300 \bibitem{Waleffe2003} F.~Waleffe, % Fabian ``Homotopy of exact coherent structures in plane shear flows'', {\em Phys. Fluids, \bf 15}, 1517 (2003). \bibitem{Faisst03} H. Faisst and B. Eckhardt, {\em Phys. Rev. Lett. \bf 91}, 224502 (2003). \bibitem{Hof03} B. Hof, A. Juel and T. Mullin, {\em Phys. Rev. Lett. \bf 91}, 244502 (2003). \bibitem{TB03} L.S.~Tuckerman and D.~Barkley, ``Stability analysis of perturbed plane Couette flow,'' {\em Phys.~Fluids \bf 11}, 1187 (1999). % 1187--1195 \bibitem{johnston04} J.-G.~Liu and H.~Johnston, ``Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term,'' {\em J. Comput. 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 \bi