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One-Dimensional Dynamics

One-dimensional dynamics has developed in the last decades into a subject in its own right. Yet, many recent results are inaccessible and have never been brought together. For this reason, we have tried to give a unified ac count of the subject and complete proofs of many results. To show what resul...

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Bibliographic Details
Main Authors: Melo, Welington de, Strien, Sebastian van (Author)
Format: eBook
Language:English
Published: Berlin, Heidelberg Springer Berlin Heidelberg 1993, 1993
Edition:1st ed. 1993
Series:Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
Subjects:
Functions Of Real Variables
Probability Theory
Real Functions
Probabilities
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 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

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