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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 nonpolynomial 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 Feigenbaumtype 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 areapreserving 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 copublished 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.

Table of Contents

Chapters

Araceli Bonifant and John Milnor — Schwarzian derivatives and cylinder maps

Holomorphic dynamics

Volodymyr Nekrashevych — Symbolic dynamics and selfsimilar 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 DíazEspinosa and Rafael de la Llave — Renormalization of arbitrary weak noises for onedimensional 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


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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 nonpolynomial 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 Feigenbaumtype 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 areapreserving 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 copublished 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 selfsimilar 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 DíazEspinosa and Rafael de la Llave — Renormalization of arbitrary weak noises for onedimensional 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