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Dynamics of transcendental entire functions with siegel disks and its applications

31 Jul 2011-Bulletin of The Korean Mathematical Society (The Korean Mathematical Society)-Vol. 48, Iss: 4, pp 713-724
TL;DR: In this paper, it was shown that the Siegel disk and its boundary correspond to those of some quadratic polynomial at the level of quasiconformality.
Abstract: We study the dynamics of transcendental entire functions with Siegel disks whose singular values are just two points. One of the two singular values is not only a superattracting fixed point with multiplicity more than two but also an asymptotic value. Another one is a critical value with free dynamics under iterations. We prove that if the multiplicity of the superattracting fixed point is large enough, then the restriction of the transcendental entire function near the Siegel point is a quadratic-like map. Therefore the Siegel disk and its boundary correspond to those of some quadratic polynomial at the level of quasiconformality. As its applications, the logarithmic lift of the above transcendental entire function has a wandering domain whose shape looks like a Siegel disk of a quadratic polynomial.

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Book
01 Jan 1995
TL;DR: In this article, Katok and Mendoza introduced the concept of asymptotic invariants for low-dimensional dynamical systems and their application in local hyperbolic theory.
Abstract: Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

3,962 citations

Book
01 Aug 2000
TL;DR: In this article, the dynamics of iterated holomorphic mappings from a Riemann surface to itself are studied, focusing on the classical case of rational maps of the RiemANN sphere.
Abstract: This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology. This third edition contains a number of minor additions and improvements: A historical survey has been added, the definition of Lattes map has been made more inclusive, and the ecalle-Voronin theory of parabolic points is described. The residu iteratif is studied, and the material on two complex variables has been expanded. Recent results on effective computability have been added, and the references have been expanded and updated. Written in his usual brilliant style, the author makes difficult mathematics look easy. This book is a very accessible source for much of what has been accomplished in the field.

1,620 citations

Book
30 May 1987
TL;DR: The Ahlfors Lectures: Acknowledgments Differentiable quasiconformal mappings The general definition Extremal geometric properties Boundary correspondence The mapping theorem Teichmuller spaces Editors' notes.
Abstract: The Ahlfors Lectures: Acknowledgments Differentiable quasiconformal mappings The general definition Extremal geometric properties Boundary correspondence The mapping theorem Teichmuller spaces Editors' notes The Additional Chapters: A supplement to Ahlfors's lectures Complex dynamics and quasiconformal mappings Hyperbolic structures on three-manifolds that fiber over the circle.

1,561 citations


"Dynamics of transcendental entire f..." refers background in this paper

  • ...(1) The Siegel disk ∆̃ of f̃ centered at the origin is a “quadratic-polynomial-type” Siegel disk....

    [...]

  • ...(ii) The above three statements (1), (2) and (3) hold for all n ≥ n0....

    [...]

  • ...(1) The inequality |fc0(z)− c0| > 3r holds on the circle ∂D(ε0, r)....

    [...]

Book
01 Jan 1993
TL;DR: In this article, the authors propose a topological theory of circle Diffeomorphisms and the Combinatorics of one-dimensional endomorphisms, based on the theory of Denjoy inequalities.
Abstract: 0. Introduction.- I. Circle Diffeomorphisms.- 1. The Combinatorial Theory of Poincare.- 2. The Topological Theory of Denjoy.- 2.a The Denjoy Inequality.- 2.b Ergodicity.- 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.- 3.a Examples.- 3.b Hofbauer's Tower Construction.- 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.- l.a Some Facts about the Schwarzian Derivative.- 2. Distortion of Cross-Ratios.- 2.a The Zygmund Conditions.- 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 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.

1,048 citations

BookDOI

972 citations


"Dynamics of transcendental entire f..." refers background in this paper

  • ...(3) The critical value fc0(c0n0/(n0 − λ)) is in the disk D(0, 2r)....

    [...]

  • ...(ii) The above three statements (1), (2) and (3) hold for all n ≥ n0....

    [...]