Fractal Dimensions for Poincare Recurrences

Preface Chapter 1. Introduction 1(6) PART I. FUNDAMENTALS 7(68) Chapter 2. Symbolic Systems 9(26) 2.1. Specified subshifts 9(3) 2.1.1 Ultrametric space 11(1) 2.2. Ordered topological Markov chains 12(5) 2.3. Multipermutative systems 17(11) 2.3.1 Polysymbolic generalization 19(1) 2.3.2 Topological conjugation of polysymbolic minimal systems 20(3) 2.3.3 Nonminimal multipermutative systems 23(5) 2.4. Topological pressure 28(7) 2.4.1 Dimension-like definition of topological pressure 32(3) Chapter 3. Geometric Constructions 35(18) 3.1. Moran constructions 35(8) 3.1.1 Generalized Moran constructions 37(3) 3.1.2 Invariant subsets of Markov maps 40(3) 3.2. Topological pressure and Hausdorff dimension 43(5) 3.2.1 Hausdorff and box dimensions 43(2) 3.2.2 Bowen's equation 45(1) 3.2.3 Moran covers 45(3) 3.3. Strong Moran construction 48(1) 3.4. Controlled packing of cylinders 48(1) 3.5. Sticky sets 49(4) 3.5.1 Geometric constructions of sticky sets 51(2) Chapter 4. The Spectrum of Dimensions for Poincar ecurrences 53(22) 4.1. Generalized Caratheodory construction 53(4) 4.1.1 Examples 54(3) 4.2. The spectrum of dimensions for recurrences 57(1) 4.3. Dimension and capacities 58(1) 4.4. The appropriate gauge functions 59(4) 4.5. General properties of the dimension for recurrences 63(2) 4.6. Dimension for minimal sets 65(12) 4.6.1 The gauge function ξ(t) 1/t 66(1) 4.6.2 Rotations of the circle 66(3) 4.6.3 Denjoy example 69(3) 4.6.4 Multidimensional rotation 72(3) PART II. ZERO-DIMENSIONAL INVARIANT SETS 75(32) Chapter 5. Uniformly Hyperbolic Repellers 77(10) 5.1. Spectrum of Lyapunov exponents 78(1) 5.2. The controlled-packing condition 79(4) 5.2.1 Proof of Lemma 5.1 80(2) 5.2.2 Proof of Lemma 5.2 82(1) 5.3. Spectra under the gap condition 83(4) Chapter 6. Non-Uniformly Hyperbolic Repellers 87(8) 6.1. No orbits in the critical set 88(2) 6.2. The critical set contains an orbit 90(5) Chapter 7. The Spectrum for a Sticky Set 95(4) 7.1. The spectrum for Poincar ecurrences 95(4) Chapter 8. Rhythmical Dynamics 99(8) 8.1. Set-up 99(1) 8.2. Dimensions for Poincar ecurrences 100(2) 8.2.1 The case of an autonomous rhythm function 100(1) 8.2.2 The case of non-autonomous rhythm function 101(1) 8.3. The spectrum of dimensions 102(7) 8.3.1 Autonomous 102(1) 8.3.2 Non-autonomous 103(4) PART III. ONE-DIMENSIONAL SYSTEMS 107(26) Chapter 9. Markov Maps of the Interval 109(8) 9.1. The spectrum of dimensions 110(7) Chapter 10. Suspended Flows 117(16) 10.1. Suspended flows over specified subshifts 117(1) 10.1.1 Poincar ecurrences 118(1) 10.1.2 Suspended flow 118(1) 10.2. Bowen Walters' distance 118(1) 10.3. Spectrum of dimensions 119(16) 10.3.1 The Poincar ecurrence 119(1) 10.3.2 The spectrum 120(1) 10.3.3 Main results 120(7) 10.3.4 Proof of Claim 10.1 127(2) 10.3.5 Proof of Claim 10.2 129(4) PART IV. MEASURE THEORETICAL RESULTS 133(40) Chapter 11. Invariant Measures and Poincar ecurrences 135(14) 11.1. Pointwise dimension and local rates 135(2) 11.2. The SMB theorem 137(1) 11.3. Kolmogorov complexity and Brudno's theorem 137(1) 11.4. The local rate of return times 138(5) 11.4.1 Proof of Theorem 11.3 based on the SMB Theorem 138(2) 11.4.2 Proof of Theorem 11.3 based on Brudno's Theorem 140(1) 11.4.3 Rotations of the circle 141(2) 11.5. Remarks on local rates 143(2) 11.6. The q-pointwise dimension 145(4) Chapter 12. Dimensions for Measures and q-Pointwise Dimension 149(18) 12.1. Preliminaries and motivation 149(2) 12.2. A formula for measures 151(2) 12.3. The q-pointwise dimension 153(3) 12.4. Sticky sets 156(5) 12.5. Remarks on the q-pointwise dimension 161(6) Chapter 13. The Variational Principle 167(6) 13.1. Preliminaries and motivation 167(4) 13.2. A variational principle for the spectrum 171(1) 13.3. The variational principle for suspended flows 172(1) PART V. PHYSICAL INTERPRETATION AND APPLICATIONS 173(42) Chapter 14. Intuitive Explanation of Some Notions and Results of this Book 175(10) 14.1. Ergodic conformal repellers 175(3) 14.1.1 Entropy 175(1) 14.1.2 Lyapunov exponents 176(1) 14.1.3 The spectrum of dimensions for Poincar ecurrences 177(1) 14.2. (Non-ergodic) Conformal repellers 178(7) 14.2.1 The entropy spectrum for Lyapunov exponents 179(1) 14.2.2 The spectrum of dimensions for Poincar ecurrences 179(2) 14.2.3 A Legendre-transform pair 181(4) Chapter 15. Poincar ecurrences in Hamiltonian Systems 185(10) 15.1. Introduction 185(1) 15.2. Asymptotic distributions 185(3) 15.3. A self-similar space-time situation 188(2) 15.4. Recurrence multifractality 190(2) 15.5. Critical exponents 192(1) 15.6. Final remarks 193(2) Chapter 16. Chaos Synchronization 195(20) 16.1. Synchronization 195(2) 16.1.1 Periodic oscillations 196(1) 16.2. Poincar ecurrences 197(4) 16.2.1 Poincar ecurrences for subsystems 198(3) 16.3. Topological synchronization 201(3) 16.4. Indicators of synchronization 204(3) 16.5. Computation of Poincar ecurrences 207(3) 16.6. Final remarks 210(5) PART VI. APPENDICES 215(20) Chapter 17. Some Known Facts about Recurrences 217(4) 17.1. Almost everyone comes back 217(2) 17.2. Kac's theorem 219(2) Chapter 18. Birkhoff's Individual Theorem 221(6) 18.1. Some general definitions 221(1) 18.2. Proof of the Birkhoff's theorem 222(5) Chapter 19. The Shannon McMillan Breiman Theorem 227(6) 19.1. Introduction 227(1) 19.2. The theorem 228(1) 19.3. Proof of the theorem 228(5) Chapter 20. Amalgamation and Fragmentation 233(2) References 235(8) Subject Index 243