Conformal Dynamics And Hyperbolic Geometry

Author : Francis Bonahon
Publisher : American Mathematical Soc.
Page : 266 pages
File Size : 43,9 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821853481

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Conformal Dynamics and Hyperbolic Geometry by Francis Bonahon Pdf

This volume contains the proceedings of the Conference on Conformal Dynamics and Hyperbolic Geometry, held October 21-23, 2010, in honor of Linda Keen's 70th birthday. This volume provides a valuable introduction to problems in conformal and hyperbolic geometry and one dimensional, conformal dynamics. It includes a classic expository article by John Milnor on the structure of hyperbolic components of the parameter space for dynamical systems arising from the iteration of polynomial maps in the complex plane. In addition there are foundational results concerning Teichmuller theory, the geometry of Fuchsian and Kleinian groups, domain convergence properties for the Poincare metric, elaboration of the theory of the universal solenoid, the geometry of dynamical systems acting on a circle, and realization of Thompson's group as a mapping class group for a uniformly asymptotically affine circle endomorphism. The portion of the volume dealing with complex dynamics will appeal to a diverse group of mathematicians. Recently many researchers working in a wide range of topics, including topology, algebraic geometry, complex analysis, and dynamical systems, have become involved in aspects of this field.

Author : Francis Bonahon
Publisher : American Mathematical Soc.
Page : 266 pages
File Size : 45,9 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821890264

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Conformal Dynamics and Hyperbolic Geometry by Francis Bonahon Pdf

Author : Francis Bonahon
Publisher : Unknown
Page : 0 pages
File Size : 45,5 Mb
Release : 2012
Category : Electronic
ISBN : 0821853481

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Conformal dynamics and hyperbolic geometry : Conference on Conformal Dynamics and Hyperbolic Geometry, in honor of Linda Keen's 70th birthday, Graduate School and University Center of CUNY, New York, NY, October 21 - 23, 2010 by Francis Bonahon Pdf

Author : John M. Mackay,Jeremy T. Tyson
Publisher : American Mathematical Soc.
Page : 162 pages
File Size : 47,8 Mb
Release : 2010
Category : Mathematics
ISBN : 9780821852293

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Conformal Dimension by John M. Mackay,Jeremy T. Tyson Pdf

Conformal dimension measures the extent to which the Hausdorff dimension of a metric space can be lowered by quasisymmetric deformations. Introduced by Pansu in 1989, this concept has proved extremely fruitful in a diverse range of areas, including geometric function theory, conformal dynamics, and geometric group theory. This survey leads the reader from the definitions and basic theory through to active research applications in geometric function theory, Gromov hyperbolic geometry, and the dynamics of rational maps, amongst other areas. It reviews the theory of dimension in metric spaces and of deformations of metric spaces. It summarizes the basic tools for estimating conformal dimension and illustrates their application to concrete problems of independent interest. Numerous examples and proofs are provided. Working from basic definitions through to current research areas, this book can be used as a guide for graduate students interested in this field, or as a helpful survey for experts. Background needed for a potential reader of the book consists of a working knowledge of real and complex analysis on the level of first- and second-year graduate courses.

Author : Tushar Das,David Simmons,Mariusz Urbański
Publisher : American Mathematical Soc.
Page : 281 pages
File Size : 48,9 Mb
Release : 2017-04-14
Category : Geometry, Hyperbolic
ISBN : 9781470434656

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Geometry and Dynamics in Gromov Hyperbolic Metric Spaces by Tushar Das,David Simmons,Mariusz Urbański Pdf

This book presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Particular emphasis is paid to the geometry of their limit sets and on behavior not found in the proper setting. The authors provide a number of examples of groups which exhibit a wide range of phenomena not to be found in the finite-dimensional theory. The book contains both introductory material to help beginners as well as new research results, and closes with a list of attractive unsolved problems.

Author : Curtis T. McMullen
Publisher : Princeton University Press
Page : 214 pages
File Size : 51,7 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400882557

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Complex Dynamics and Renormalization (AM-135), Volume 135 by Curtis T. McMullen Pdf

Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics. Its central concern is the structure of an infinitely renormalizable quadratic polynomial f(z) = z2 + c. As discovered by Feigenbaum, such a mapping exhibits a repetition of form at infinitely many scales. Drawing on universal estimates in hyperbolic geometry, this work gives an analysis of the limiting forms that can occur and develops a rigidity criterion for the polynomial f. This criterion supports general conjectures about the behavior of rational maps and the structure of the Mandelbrot set. The course of the main argument entails many facets of modern complex dynamics. Included are foundational results in geometric function theory, quasiconformal mappings, and hyperbolic geometry. Most of the tools are discussed in the setting of general polynomials and rational maps.

Author : Michel Coornaert,Athanase Papadopoulos
Publisher : Springer
Page : 145 pages
File Size : 45,5 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540475736

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Symbolic Dynamics and Hyperbolic Groups by Michel Coornaert,Athanase Papadopoulos Pdf

Gromov's theory of hyperbolic groups have had a big impact in combinatorial group theory and has deep connections with many branches of mathematics suchdifferential geometry, representation theory, ergodic theory and dynamical systems. This book is an elaboration on some ideas of Gromov on hyperbolic spaces and hyperbolic groups in relation with symbolic dynamics. Particular attention is paid to the dynamical system defined by the action of a hyperbolic group on its boundary. The boundary is most oftenchaotic both as a topological space and as a dynamical system, and a description of this boundary and the action is given in terms of subshifts of finite type. The book is self-contained and includes two introductory chapters, one on Gromov's hyperbolic geometry and the other one on symbolic dynamics. It is intended for students and researchers in geometry and in dynamical systems, and can be used asthe basis for a graduate course on these subjects.

Author : Lennart Carleson,Theodore W. Gamelin
Publisher : Springer Science & Business Media
Page : 181 pages
File Size : 53,9 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9781461243649

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Complex Dynamics by Lennart Carleson,Theodore W. Gamelin Pdf

A discussion of the properties of conformal mappings in the complex plane, closely related to the study of fractals and chaos. Indeed, the book ends in a detailed study of the famous Mandelbrot set, which describes very general properties of such mappings. Focusing on the analytic side of this contemporary subject, the text was developed from a course taught over several semesters and aims to help students and instructors to familiarize themselves with complex dynamics. Topics covered include: conformal and quasi-conformal mappings, fixed points and conjugations, basic rational iteration, classification of periodic components, critical points and expanding maps, some applications of conformal mappings, the local geometry of the Fatou set, and quadratic polynomials and the Mandelbrot set.

Author : Vadim A. Kaimanovich,Mikhail Lyubich
Publisher : American Mathematical Soc.
Page : 134 pages
File Size : 55,9 Mb
Release : 2005
Category : Complex manifolds
ISBN : 9780821836156

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Conformal and Harmonic Measures on Laminations Associated with Rational Maps by Vadim A. Kaimanovich,Mikhail Lyubich Pdf

This book is dedicated to Dennis Sullivan on the occasion of his 60th birthday. The framework of affine and hyperbolic laminations provides a unifying foundation for many aspects of conformal dynamics and hyperbolic geometry. The central objects of this approach are an affine Riemann surface lamination $\mathcal A$ and the associated hyperbolic 3-lamination $\mathcal H$ endowed with an action of a discrete group of isomorphisms. This action is properly discontinuous on $\mathcal H$, which allows one to pass to the quotient hyperbolic lamination $\mathcal M$. Our work explores natural ``geometric'' measures on these laminations. We begin with a brief self-contained introduction to the measure theory on laminations by discussing the relationship between leafwise, transverse and global measures. The central themes of our study are: leafwise and transverse ``conformal streams'' on an affine lamination $\mathcal A$ (analogues of the Patterson-Sullivan conformal measures for Kleinian groups), harmonic and invariant measures on the corresponding hyperbolic lamination $\mathcal H$, the ``Anosov--Sinai cocycle'', the corresponding ``basic cohomology class'' on $\mathcal A$ (which provides an obstruction to flatness), and the Busemann cocycle on $\mathcal H$. A number of related geometric objects on laminations -- in particular, the backward and forward Poincare series and the associated critical exponents, the curvature forms and the Euler class, currents and transverse invariant measures, $\lambda$-harmonic functions and the leafwise Brownian motion -- are discussed along the lines. The main examples are provided by the laminations arising from the Kleinian and the rational dynamics. In the former case, $\mathcal M$ is a sublamination of the unit tangent bundle of a hyperbolic 3-manifold, its transversals can be identified with the limit set of the Kleinian group, and we show how the classical theory of Patterson-Sullivan measures can be recast in terms of our general approach. In the latter case, the laminations were recently constructed by Lyubich and Minsky in [LM97]. Assuming that they are locally compact, we construct a transverse $\delta$-conformal stream on $\mathcal A$ and the corresponding $\lambda$-harmonic measure on $\mathcal M$, where $\lambda=\delta(\delta-2)$. We prove that the exponent $\delta$ of the stream does not exceed 2 and that the affine laminations are never flat except for several explicit special cases (rational functions with parabolic Thurston orbifold).

Author : Bodil Branner,Núria Fagella
Publisher : Cambridge University Press
Page : 433 pages
File Size : 53,6 Mb
Release : 2014-01-23
Category : Mathematics
ISBN : 9781107042919

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Quasiconformal Surgery in Holomorphic Dynamics by Bodil Branner,Núria Fagella Pdf

A comprehensive introduction to quasiconformal surgery in holomorphic dynamics. Contains a wide variety of applications and illustrations.

Author : Christian Bonatti,Lorenzo J. Díaz,Marcelo Viana
Publisher : Springer Science & Business Media
Page : 408 pages
File Size : 48,8 Mb
Release : 2004-09-30
Category : Mathematics
ISBN : 3540220666

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Dynamics Beyond Uniform Hyperbolicity by Christian Bonatti,Lorenzo J. Díaz,Marcelo Viana Pdf

The notion of uniform hyperbolicity, introduced by Steve Smale in the early sixties, unified important developments and led to a remarkably successful theory for a large class of systems: uniformly hyperbolic systems often exhibit complicated evolution which, nevertheless, is now rather well understood, both geometrically and statistically. Another revolution has been taking place in the last couple of decades, as one tries to build a global theory for "most" dynamical systems, recovering as much as possible of the conclusions of the uniformly hyperbolic case, in great generality. This book aims to put such recent developments in a unified perspective, and to point out open problems and likely directions for further progress. It is aimed at researchers, both young and senior, willing to get a quick, yet broad, view of this part of dynamics. Main ideas, methods, and results are discussed, at variable degrees of depth, with references to the original works for details and complementary information.

Author : Marc Burger,Alessandra Iozzi
Publisher : Springer Science & Business Media
Page : 494 pages
File Size : 47,9 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662047439

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Rigidity in Dynamics and Geometry by Marc Burger,Alessandra Iozzi Pdf

This volume of proceedings is an offspring of the special semester Ergodic Theory, Geometric Rigidity and Number Theory which was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK, from Jan uary until July, 2000. Beside the activities during the semester, there were workshops held in January, March and July, the first being of introductory nature with five short courses delivered over a week. Although the quality of the workshops was excellent throughout the semester, the idea of these proceedings came about during the March workshop, which is hence more prominently represented, The format of the volume has undergone many changes, but what has remained untouched is the enthusiasm of the contributors since the onset of the project: suffice it to say that even though only two months elapsed between the time we contacted the potential authors and the deadline to submit the papers, the deadline was respected in the vast majority of the cases. The scope of the papers is not completely uniform throughout the volume, although there are some points in common. We asked the authors to write papers keeping in mind the idea that they should be accessible to students. At the same time, we wanted the papers not to be a summary of results that appeared somewhere else.

Author : Curtis T. McMullen
Publisher : Unknown
Page : 214 pages
File Size : 41,8 Mb
Release : 1994
Category : Science
ISBN : 0691029822

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Complex Dynamics and Renormalization by Curtis T. McMullen Pdf

Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics. Its central concern is the structure of an infinitely renormalizable quadratic polynomial f(z) = z2 + c. As discovered by Feigenbaum, such a mapping exhibits a repetition of form at infinitely many scales. Drawing on universal estimates in hyperbolic geometry, this work gives an analysis of the limiting forms that can occur and develops a rigidity criterion for the polynomial f. This criterion supports general conjectures about the behavior of rational maps and the structure of the Mandelbrot set. The course of the main argument entails many facets of modern complex dynamics. Included are foundational results in geometric function theory, quasiconformal mappings, and hyperbolic geometry. Most of the tools are discussed in the setting of general polynomials and rational maps.

Author : Araceli Bonifant,Misha Lyubich,Scott Sutherland
Publisher : Princeton University Press
Page : 799 pages
File Size : 45,7 Mb
Release : 2014-03-16
Category : Mathematics
ISBN : 9780691159294

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Frontiers in Complex Dynamics by Araceli Bonifant,Misha Lyubich,Scott Sutherland Pdf

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing. This collection will be useful to students and researchers for decades to come. The contributors are Marco Abate, Marco Arizzi, Alexander Blokh, Thierry Bousch, Xavier Buff, Serge Cantat, Tao Chen, Robert Devaney, Alexandre Dezotti, Tien-Cuong Dinh, Romain Dujardin, Hugo García-Compeán, William Goldman, Rotislav Grigorchuk, John Hubbard, Yunping Jiang, Linda Keen, Jan Kiwi, Genadi Levin, Daniel Meyer, John Milnor, Carlos Moreira, Vincente Muñoz, Viet-Anh Nguyên, Lex Oversteegen, Ricardo Pérez-Marco, Ross Ptacek, Jasmin Raissy, Pascale Roesch, Roberto Santos-Silva, Dierk Schleicher, Nessim Sibony, Daniel Smania, Tan Lei, William Thurston, Vladlen Timorin, Sebastian van Strien, and Alberto Verjovsky.

Author : Yunping Jiang
Publisher : World Scientific
Page : 327 pages
File Size : 40,7 Mb
Release : 1996-09-20
Category : Science
ISBN : 9789814500173

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Renormalization And Geometry In One-dimensional And Complex Dynamics by Yunping Jiang Pdf

About one and a half decades ago, Feigenbaum observed that bifurcations, from simple dynamics to complicated ones, in a family of folding mappings like quadratic polynomials follow a universal rule (Coullet and Tresser did some similar observation independently). This observation opened a new way to understanding transition from nonchaotic systems to chaotic or turbulent system in fluid dynamics and many other areas. The renormalization was used to explain this observed universality. This research monograph is intended to bring the reader to the frontier of this active research area which is concerned with renormalization and rigidity in real and complex one-dimensional dynamics. The research work of the author in the past several years will be included in this book. Most recent results and techniques developed by Sullivan and others for an understanding of this universality as well as the most basic and important techniques in the study of real and complex one-dimensional dynamics will also be included here.