One-Dimensional Dynamics

✍ Scribed by Welington De Melo, Sebastian Van Strien

• Publisher: Springer
• Year: 1993
• Tongue: English
• Leaves: 594
• Series: Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge
• Edition: 1ST
• Category: Library

✦ Synopsis

This monograph gives an account of the state of the art in one-dimensional dynamical systems. It presents the theory in a unified way emphasizing the similarities and differences between invertible and non-invertible dynamics (i.e., between diffeomorphisms and endomorphisms).It starts with the invertible case: the combinatorial topological, ergodic and smooth structures are analysed extensively. Then it proceeds by showing that endomorphisms have a much richer dynamics, but that the theory for these endomorphisms can still be developed along the same lines and with similar tools. Moreover, holomorphic dynamical systems are shown to be based on similar principles. In fact, it is shown that complex analytic tools are very powerful even for the study of real one-dimensional systems. Several results in this book are new. Moreover, the exciting new developments on universality and renormalization due to D. Sullivan, are presented here in full detail for the first time.

✦ Table of Contents

Introduction......Page 9
Circle Diffeomorphisms......Page 22
The Combinatorial Theory of Poincaré......Page 24
The Topological Theory of Denjoy......Page 43
The Denjoy Inequality......Page 54
Ergodicity......Page 55
Smooth Conjugacy Results......Page 56
Families of Circle Diffeomorphisms; Arnol'd Tongues......Page 73
Counter-Examples to Smooth Linearizability......Page 77
Frequency of Smooth Linearizability in Families......Page 81
Some Historical Comments and Further Remarks......Page 83
The Combinatorics of Endomorphisms......Page 85
The Theorem of Sarkovskii......Page 86
Covering Maps of the Circle as Dynamical Systems......Page 92
The Kneading Theory and Combinatorial Equivalence......Page 96
Examples......Page 111
Hofbauer's Tower Construction......Page 113
Full Families and Realization of Maps......Page 119
Families of Maps and Renormalization......Page 141
Piecewise Monotone Maps can be Modelled by Polynomial Maps......Page 156
The Topological Entropy......Page 165
The Piecewise Linear Model......Page 173
Continuity of the Topological Entropy......Page 187
Monotonicity of the Kneading Invariant for the Quadratic Family......Page 195
Some Historical Comments and Further Remarks......Page 198
Structural Stability and Hyperbolicity......Page 202
The Dynamics of Rational Mappings......Page 203
Structural Stability and Hyperbolicity......Page 217
Hyperbolicity in Maps with Negative Schwarzian Derivative......Page 231
The Structure of the Non-Wandering Set......Page 236
Hyperbolicity in Smooth Maps......Page 247
Misiurewicz Maps are Almost Hyperbolic......Page 258
Some Further Remarks and Open Questions......Page 264
The Structure of Smooth Maps......Page 267
The Cross-Ratio: the Minimum and Koebe Principle......Page 271
Some Facts about the Schwarzian Derivative......Page 281
Distortion of Cross-Ratios......Page 284
The Zygmund Conditions......Page 290
Koebe Principles on Iterates......Page 293
Some Simplifications and the Induction Assumption......Page 300
The Pullback of Space: the Koebe/Contraction Principle......Page 303
Disjointness of Orbits of Intervals......Page 306
Wandering Intervals Accumulate on Turning Points......Page 310
Topological Properties of a Unimodal Pullback......Page 313
The Non-Existence of Wandering Intervals......Page 317
Finiteness of Attractors......Page 319
Some Further Remarks and Open Questions......Page 323
Ergodic Properties and Invariant Measures......Page 325
Ergodicity, Attractors and Bowen-Ruelle-Sinai Measures......Page 327
Invariant Measures for Markov Maps......Page 349
Constructing Invariant Measures by Inducing......Page 359
Constructing Invariant Measures by Pulling Back......Page 372
Transitive Maps Without Finite Continuous Measures......Page 388
Jakobson's Theorem......Page 396
Some Further Remarks and Open Questions......Page 426
Renormalization......Page 430
The Renormalization Operator......Page 431
The Real Bounds......Page 446
Bounded Geometry......Page 454
The Pullback Argument......Page 458
The Complex Bounds......Page 474
Riemann Surface Laminations......Page 491
The Almost Geodesic Principle......Page 515
Renormalization is Contracting......Page 524
Universality of the Attracting Cantor Set......Page 536
Some Further Remarks and Open Questions......Page 542
Some Terminology in Dynamical Systems......Page 545
Some Background in Topology......Page 546
Some Results from Analysis and Measure Theory......Page 548
Some Results from Ergodic Theory......Page 550
Some Background in Complex Analysis......Page 551
Some Results from Functional Analysis......Page 570
Bibliography......Page 571