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One-dimensional dynamics

Author: Welington de Melo; Sebastian van Strien
Publisher: Berlin ; New York : Springer-Verlag, ©1993.
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 25.
Edition/Format:   Print book : Document   Computer File : EnglishView all editions and formats
Summary:

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. It includes coverage of new  Read more...

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Additional Physical Format: Online version:
Melo, Welington de.
One-dimensional dynamics.
Berlin ; New York : Springer-Verlag, ©1993
(OCoLC)680651941
Material Type: Document, Internet resource
Document Type: Book, Computer File, Internet Resource
All Authors / Contributors: Welington de Melo; Sebastian van Strien
ISBN: 3540564128 9783540564126 0387564128 9780387564128 9783642780455 3642780458
OCLC Number: 27938327
Description: xiii, 605 pages : illustrations ; 25 cm.
Contents: Ch. 0. Introduction --
Ch. I. Circle Diffeomorphisms --
1. The Combinatorial Theory of Poincare --
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 --
Ch. 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 --
Ch. 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 --
Ch. 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 --
Ch. 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 --
Ch. 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 --
Ch. 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.
Series Title: Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 25.
Responsibility: Welington de Melo, Sebastian van Strien.
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