Author | Melo, Welington de. author |
---|---|
Title | One-Dimensional Dynamics [electronic resource] / by Welington de Melo, Sebastian van Strien |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1993 |
Connect to | http://dx.doi.org/10.1007/978-3-642-78043-1 |
Descript | XIV, 606 p. online resource |
0. Introduction -- I. Circle Diffeomorphisms -- 1. The Combinatorial Theory of Poincarรฉ -- 2. The Topological Theory of Denjoy -- 3. Smooth Conjugacy Results -- 4. Families of Circle Diffeomorphisms; Arnolโd 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. Some Further 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 Jakobsonโs 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