Rigidity and inflexibility in conformal dynamics
type: Item
instance of: scholarly article
title: Rigidity and inflexibility in conformal dynamics
zbMATH Open document ID: 0914.58015
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/225214
full work available at URL: McMullen.MAN.html
review text: The author presents a connection between the rigidity of hyperbolic 3-manifolds and universal scaling phenomena in dynamics. Starting with an inflexibility theorem for 3-manifolds of infinite volume and the notion of generalized Mostow rigidity, the author connects this inflexibility to dynamics. In Sections 2-6, the author discusses the geometrization of 3-manifolds which fiber over the circle; the extension of the inflexibility of Kleinian groups and their limit sets to certain other conformal dynamical systems and their Julia sets; the renormalization of unimodal maps; critical circle maps; and the self-similarity of Siegel disks. In Section 7, the author concludes with progress towards the classification of hyperbolic manifolds. The paper also contains a useful table which summarizes the parallels emerging between hyperbolic manifolds, quadratic-like maps on the interval, critical circle maps and Siegel disks.
Mathematics Subject Classification ID: 37E99
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 37F99
zbMATH DE Number: 1184293
zbMATH Keywords: rigidity of hyperbolic 3-manifolds
zbMATH Keywords: universal scaling
zbMATH Keywords: renormalization of unimodal maps
zbMATH Keywords: real-analytic circle homeomorphisms with critical points
zbMATH Keywords: self-similarity of Siegel disks
zbMATH Keywords: inflexibility
zbMATH Keywords: circle maps
author: Curtis T. McMullen
reviewed by: Zhuan Ye
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1184293