Fractal Dimensions for Poincare Recurrences

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Juni 2006



This book is devoted to an important branch of the dynamical systems theory : the study of the fine (fractal) structure of Poincare recurrences -instants of time when the system almost repeats its initial state. The authors were able to write an entirely self-contained text including many insights and examples, as well as providing complete details of proofs. The only prerequisites are a basic knowledge of analysis and topology. Thus this book can serve as a graduate text or self-study guide for courses in applied mathematics or nonlinear dynamics (in the natural sciences). Moreover, the book can be used by specialists in applied nonlinear dynamics following the way in the book. The authors applied the mathematical theory developed in the book to two important problems: distribution of Poincare recurrences for nonpurely chaotic Hamiltonian systems and indication of synchronization regimes in coupled chaotic individual systems.

* Portions of the book were published in an article that won the title "month's new hot paper in the field of Mathematics" in May 2004
* Rigorous mathematical theory is combined with important physical applications
* Presents rules for immediate action to study mathematical models of real systems
* Contains standard theorems of dynamical systems theory


1;Cover ;1 2;Preface;6 3;Contents;8 4;Introduction;14 5;Fundamentals;20 5.1;Symbolic Systems;22 5.1.1;Specified subshifts;22;Ultrametric space;24 5.1.2;Ordered topological Markov chains;25 5.1.3;Multipermutative systems;30;Polysymbolic generalization;32;Topological conjugation of polysymbolic minimal systems;33;Nonminimal multipermutative systems;36 5.1.4;Topological pressure;41;Dimension-like definition of topological pressure;45 5.2;Geometric Constructions;48 5.2.1;Moran constructions;48;Generalized Moran constructions;50;Invariant subsets of Markov maps;53 5.2.2;Topological pressure and Hausdorff dimension;56;Hausdorff and box dimensions;56;Bowen's equation;58;Moran covers;58 5.2.3;Strong Moran construction;61 5.2.4;Controlled packing of cylinders;61 5.2.5;Sticky sets;62;Geometric constructions of sticky sets;64 5.3;The Spectrum of Dimensions for Poincaré Recurrences;66 5.3.1;Generalized Carathéodory construction;66;Examples (see Table 4.3);67 5.3.2;The spectrum of dimensions for recurrences;70 5.3.3;Dimension and capacities;71 5.3.4;The appropriate gauge functions;72 5.3.5;General properties of the dimension for recurrences;76 5.3.6;Dimension for minimal sets;78;The gauge function xi(t) =1/t;79;Rotations of the circle;79;Denjoy example;82;Multidimensional rotation;85 6;Zero-Dimensional Invariant Sets;88 6.1;Uniformly Hyperbolic Repellers;90 6.1.1;Spectrum of Lyapunov exponents;91 6.1.2;The controlled-packing condition;92;Proof of Lemma 5.1;93;Proof of Lemma 5.2;95 6.1.3;Spectra under the gap condition;96 6.2;Non-Uniformly Hyperbolic Repellers;100 6.2.1;No orbits in the critical set;101 6.2.2;The critical set contains an orbit;103 6.3;The Spectrum for a Sticky Set;108 6.3.1;The spectrum for Poincaré recurrences;108 6.4;Rhythmical Dynamics;112 6.4.1;Set-up;112 6.4.2;Dimensions for Poincaré recurrences;113;The case
of an autonomous rhythm function phi;113;The case of non-autonomous rhythm function phi;114 6.4.3;The spectrum of dimensions;115;Autonomous phi;115;Non-autonomous phi;116 7;One-Dimensional Systems;120 7.1;Markov Maps of the Interval;122 7.1.1;The spectrum of dimensions;123 7.2;Suspended Flows;130 7.2.1;Suspended flows over specified subshifts;130;Poincaré recurrences;131;Suspended flow;131 7.2.2;Bowen-Walters' distance;131 7.2.3;Spectrum of dimensions;132;The Poincaré recurrence;132;The spectrum;133;Main results;133;Proof of Claim 10.1;140;Proof of Claim 10.2;142 8;Measure Theoretical Results;146 8.1;Invariant Measures and Poincaré Recurrences;148 8.1.1;Pointwise dimension and local rates;148 8.1.2;The SMB theorem;150 8.1.3;Kolmogorov complexity and Brudno's theorem;150 8.1.4;The local rate of return times;151;Proof of Theorem 11.3 based on the SMB Theorem;151;Proof of Theorem 11.3 based on Brudno's Theorem;153;Rotations of the circle;154 8.1.5;Remarks on local rates;156 8.1.6;The q-pointwise dimension;158 8.2;Dimensions for Measures and q-Pointwise Dimension;162 8.2.1;Preliminaries and motivation;162 8.2.2;A formula for measures;164 8.2.3;The q-pointwise dimension;166 8.2.4;Sticky sets;169 8.2.5;Remarks on the q-pointwise dimension;174 8.3;The Variational Principle;180 8.3.1;Preliminaries and motivation;180 8.3.2;A variational principle for the spectrum;184 8.3.3;The variational principle for suspended flows;185 9;Physical Interpretation and Applications;186 9.1;Intuitive Explanation of Some Notions and Results of this Book;188 9.1.1;Ergodic conformal repellers;188;Entropy;188;Lyapunov exponents;189;The spectrum of dimensions for Poincaré recurrences;190 9.1.2;(Non-ergodic) Conformal repellers;191;The entropy spectrum for Lyapunov exponents;192;The spectrum of dimensions for Poincaré recurrences;192;A Legendre-trans
form pair;194 9.2;Poincaré Recurrences in Hamiltonian Systems;198 9.2.1;Introduction;198 9.2.2;Asymptotic distributions;198 9.2.3;A self-similar space-time situation;201 9.2.4;Recurrence multifractality;203 9.2.5;Critical exponents;205 9.2.6;Final remarks;206 9.3;Chaos Synchronization;208 9.3.1;Synchronization;208;Periodic oscillations;209 9.3.2;Poincaré recurrences;210;Poincaré recurrences for subsystems;211 9.3.3;Topological synchronization;214 9.3.4;Indicators of synchronization;217 9.3.5;Computation of Poincaré recurrences;220 9.3.6;Final remarks;223 10;Appendices;228 10.1;Some Known Facts about Recurrences;230 10.1.1;Almost everyone comes back;230 10.1.2;Kac's theorem;232 10.2;Birkhoff's Individual Theorem;234 10.2.1;Some general definitions;234 10.2.2;Proof of the Birkhoff's theorem;235 10.3;The Shannon-McMillan-Breiman Theorem;240 10.3.1;Introduction;240 10.3.2;The theorem;241 10.3.3;Proof of the theorem;241 10.4;Amalgamation and Fragmentation;246 11;References;248 12;Subject Index;256


The authors started to work on the subject in 1997 because of requirements in nonlinear dynamics to find out quantities that could measure different behavior in time in dynamical systems. They introduced and studied fractal dimensions for Poincare recurrences that appeared to be new, useful characteristics of complexity of dynamics.
EAN: 9780080462394
Untertitel: 200:Adobe eBook. Sprache: Englisch. Dateigröße in MByte: 2.
Verlag: Elsevier Science
Erscheinungsdatum: Juni 2006
Seitenanzahl: 258 Seiten
Format: pdf eBook
Kopierschutz: Adobe DRM
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