This monograph gives an account of the state of the art in one-dimensional dynamical systems. The subject is studied from a combinatorial, continuous, ergodic and smooth point of view. 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. The results are presented in a unified way and with complete and thorough proofs. The study of circle maps, interval and holomorphic maps of the Riemann sphere are all shown to be based on similar principles. With this book, the reader is able to quickly get to the frontier of this exciting subject without studying many inaccessible papers.

Inhalt

0. Introduction.- I. Circle Diffeomorphisms.- 1. The Combinatorial Theory of Poincaré.- 2. The Topological Theory of Denjoy.- 3. Smooth Conjugacy Results.- 4. Families of Circle Diffeomorphisms; Arnol'd tongues.- 5. Counter-Examples to Smooth Linearizability.- 6. Frequency of Smooth Linearizability in Families.- 7. Some Historical Comments and Further Remarks.- II. The Combinatorics of One-Dimensional Endomorphisms.- 1. The Theorem of Sarkovskii.- 2. Covering Maps of the Circle as Dynamical Systems.- 3. The Kneading Theory and Combinatorial Equivalence.- 4. Full Families and Realization of Maps.- 5. Families of Maps and Renormalization.- 6. Piecewise Monotone Maps can be Modelled by Polynomial Maps.- 7. The Topological Entropy.- 8. The Piecewise Linear Model.- 9. Continuity of the Topological Entropy.- 10. Monotonicity of the Kneading Invariant for the Quadratic Family.- 11. Some Historical Comments and Further Remarks.- III. Structural Stability and Hyperbolicity.- 1. The Dynamics of Rational Mappings.- 2. Structural Stability and Hyperbolicity.- 3. Hyperbolicity in Maps with Negative Schwarzian Derivative.- 4. The Structure of the Non-Wandering Set.- 5. Hyperbolicity in Smooth Maps.- 6. Misiurewicz Maps are Almost Hyperbolic.- 7. Some Further Remarks and Open Questions.- IV. The Structure of Smooth Maps.- 1. The Cross-Ratio: the Minimum and Koebe Principle.- 2. Distortion of Cross-Ratios.- 3. Koebe Principles on Iterates.- 4. Some Simplifications and the Induction Assumption.- 5. The Pullback of Space: the Koebe/Contraction Principle.- 6. Disjointness of Orbits of Intervals.- 7. Wandering Intervals Accumulate on Turning Points.- 8. Topological Properties of a Unimodal Pullback.- 9. The Non-Existence of Wandering Intervals.- 10. Finiteness of Attractors.- 11. SomeFurther Remarks and Open Questions.- V. Ergodic Properties and Invariant Measures.- 1. Ergodicity, Attractors and Bowen-Ruelle-Sinai Measures.- 2. Invariant Measures for Markov Maps.- 3. Constructing Invariant Measures by Inducing.- 4. Constructing Invariant Measures by Pulling Back.- 5. Transitive Maps Without Finite Continuous Measures.- 6. Frequency of Maps with Positive Liapounov Exponents in Families and Jakobson's Theorem.- 7. Some Further Remarks and Open Questions.- VI. Renormalization.- 1. The Renormalization Operator.- 2. The Real Bounds.- 3. Bounded Geometry.- 4. The PullBack Argument.- 5. The Complex Bounds.- 6. Riemann Surface Laminations.- 7. The Almost Geodesic Principle.- 8. Renormalization is Contracting.- 9. Universality of the Attracting Cantor Set.- 10. Some Further Remarks and Open Questions.- VII. Appendix.- 1. Some Terminology in Dynamical Systems.- 2. Some Background in Topology.- 3. Some Results from Analysis and Measure Theory.- 4. Some Results from Ergodic Theory.- 5. Some Background in Complex Analysis.- 6. Some Results from Functional Analysis.