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Book DetailsFields Institute CommunicationsVolume: 53; 2008; 395 ppMSC: Primary 20; 37
The papers collected in this volume reflect some of the directions of research in two closely related fields: Complex Dynamics and Renormalization in Dynamical Systems.
While dynamics of polynomial mappings, particularly quadratics, has by now reached a mature state of development, much less is known about non-polynomial rational maps. The reader will be introduced into this fascinating world and a related area of transcendental dynamics by the papers in this volume. A graduate student will find an area rich with open problems and beautiful computer simulations.
A survey by V. Nekrashevych introduces the reader to iterated monodromy groups of rational mappings, a recently developed subject that links geometric group theory to combinatorics of rational maps. In this new language, many questions related to Thurston's theory of branched coverings of the sphere can be answered explicitly.
Renormalization theory occupies a central place in modern Complex Dynamics. The progress in understanding the structure of the Mandelbrot set, polynomial Julia sets, and Feigenbaum-type universalities stems from renormalization techniques. Renormalization of circle maps and rotation domains, such as Siegel disks, can be understood in the context of the classical KAM theory. Corresponding phenomena in higher dimensions, such as universal scaling in area-preserving maps in 2D, on the boundary of KAM, pose a challenging problem. A survey by H. Koch and several other papers in the volume will introduce the reader to this direction of study.
Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).
ReadershipGraduate students and research mathematicians interested in complex dynamical systems and renormalization in dynamics.
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Table of Contents
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Chapters
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Araceli Bonifant and John Milnor — Schwarzian derivatives and cylinder maps
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Holomorphic dynamics
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Volodymyr Nekrashevych — Symbolic dynamics and self-similar groups
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Douglas Childers — Are there critical points on the boundaries of mother hedgehogs?
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Laura DeMarco — Finiteness for degenerate polynomials
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Robert Devaney — Cantor webs in the parameter and dynamical planes of rational maps
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Alexey Glutsyuk — Simple proofs of uniformization theorems
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Carsten Petersen and Pascale Roesch — The Yoccoz combinatorial analytic invariant
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Lasse Rempe and Dierk Schleicher — Bifurcation loci of exponential maps and quadratic polynomials: Local connectivity, triviality of fibers, and density of hyperbolicity
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Johannes Rückert — Rational and transcendental Newton maps
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Dierk Schleicher — Newton’s method as a dynamical system: Efficient root finding of polynomials and the Riemann $\zeta $ function
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Vladlen Timorin — The external boundary of $M_2$
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Renormalization
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Hans Koch — Renormalization of vector fields
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Oliver Díaz-Espinosa and Rafael de la Llave — Renormalization of arbitrary weak noises for one-dimensional critical dynamical systems: Summary of results and numerical explorations
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Hakan Eliasson and Sergei Kuksin — KAM for the nonlinear Schrödinger equation—A short presentation
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Michael Yampolsky — Siegel disks and renormalization fixed points
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Additional Material
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RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
- Book Details
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The papers collected in this volume reflect some of the directions of research in two closely related fields: Complex Dynamics and Renormalization in Dynamical Systems.
While dynamics of polynomial mappings, particularly quadratics, has by now reached a mature state of development, much less is known about non-polynomial rational maps. The reader will be introduced into this fascinating world and a related area of transcendental dynamics by the papers in this volume. A graduate student will find an area rich with open problems and beautiful computer simulations.
A survey by V. Nekrashevych introduces the reader to iterated monodromy groups of rational mappings, a recently developed subject that links geometric group theory to combinatorics of rational maps. In this new language, many questions related to Thurston's theory of branched coverings of the sphere can be answered explicitly.
Renormalization theory occupies a central place in modern Complex Dynamics. The progress in understanding the structure of the Mandelbrot set, polynomial Julia sets, and Feigenbaum-type universalities stems from renormalization techniques. Renormalization of circle maps and rotation domains, such as Siegel disks, can be understood in the context of the classical KAM theory. Corresponding phenomena in higher dimensions, such as universal scaling in area-preserving maps in 2D, on the boundary of KAM, pose a challenging problem. A survey by H. Koch and several other papers in the volume will introduce the reader to this direction of study.
Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).
Graduate students and research mathematicians interested in complex dynamical systems and renormalization in dynamics.
-
Chapters
-
Araceli Bonifant and John Milnor — Schwarzian derivatives and cylinder maps
-
Holomorphic dynamics
-
Volodymyr Nekrashevych — Symbolic dynamics and self-similar groups
-
Douglas Childers — Are there critical points on the boundaries of mother hedgehogs?
-
Laura DeMarco — Finiteness for degenerate polynomials
-
Robert Devaney — Cantor webs in the parameter and dynamical planes of rational maps
-
Alexey Glutsyuk — Simple proofs of uniformization theorems
-
Carsten Petersen and Pascale Roesch — The Yoccoz combinatorial analytic invariant
-
Lasse Rempe and Dierk Schleicher — Bifurcation loci of exponential maps and quadratic polynomials: Local connectivity, triviality of fibers, and density of hyperbolicity
-
Johannes Rückert — Rational and transcendental Newton maps
-
Dierk Schleicher — Newton’s method as a dynamical system: Efficient root finding of polynomials and the Riemann $\zeta $ function
-
Vladlen Timorin — The external boundary of $M_2$
-
Renormalization
-
Hans Koch — Renormalization of vector fields
-
Oliver Díaz-Espinosa and Rafael de la Llave — Renormalization of arbitrary weak noises for one-dimensional critical dynamical systems: Summary of results and numerical explorations
-
Hakan Eliasson and Sergei Kuksin — KAM for the nonlinear Schrödinger equation—A short presentation
-
Michael Yampolsky — Siegel disks and renormalization fixed points