Forty years of unimodal dynamics: on the occasion of Artur Avila winning the Brin Prize
type: Item
instance of: scholarly article
title: Forty years of unimodal dynamics: on the occasion of Artur Avila winning the Brin Prize
zbMATH Open document ID: 1280.37036
author: Mikhail Lyubich
published in: Journal of Modern Dynamics
publication date: 2012-10-16T00:00:00Z
reviewed by: Alexey A. Glutsyuk
Mathematics Subject Classification ID: 37E05
Mathematics Subject Classification ID: 37E20
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37F35
Mathematics Subject Classification ID: 37F45
Mathematics Subject Classification ID: 37F50
Mathematics Subject Classification ID: 37D20
Mathematics Subject Classification ID: 37D25
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 37C40
Mathematics Subject Classification ID: 37C45
Mathematics Subject Classification ID: 37-02
zbMATH DE Number: 6094023
zbMATH Keywords: unimodal map
zbMATH Keywords: quadratic family
zbMATH Keywords: kneading sequence
zbMATH Keywords: renormalization
zbMATH Keywords: universality
zbMATH Keywords: hyperbolic map
zbMATH Keywords: stochastic map
zbMATH Keywords: complex dynamics
zbMATH Keywords: Julia set
zbMATH Keywords: Mandelbrot set
zbMATH Keywords: quadratic-like map
zbMATH Keywords: hybrid class
zbMATH Keywords: Feigenbaum map
Wikidata QID: Q56602639
MaRDI profile type: Publication
full work available at URL: jmd.2012.6.183
OpenAlex ID: W2334964429
DOI: 10.3934/JMD.2012.6.183
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review text: In 2011 Artur Avila won the Brin Prize for his fundamental contributions in three domains of dynamical systems: one-dimensional dynamics; quasi-periodic Schrödinger operators; Teichmüller dynamics and interval exchange transformations.NEWLINENEWLINEThe paper under review is written on the occasion of his prize winning. It is an excellent survey of the development of one-dimensional dynamics from 1970s till nowadays, when real analytic unimodal dynamics has became fully probabilistically understood. Its most part concerns the unimodal dynamics (dynamics of a mapping of a segment to itself with one critical point that is an extremum). It makes an emphasis on works of Avila, who has been the key player in the final stage of the story.NEWLINENEWLINEThe paper is very well-written and can be read by non-specialists: all the definitions and basic results are recalled, and key ideas of proofs are explained. It starts with the basic theory: Milnor-Thurston kneading combinatorial invariant, Feigenbaum map and universality phenomenon, stochastic maps (i.e., having an absolutely continuous invariant probability measure) and their abundance (Jakobson's theorem). The next sections give a survey of the development of complex methods: famous conjectures on density of hyperbolicity in quadratic family and local connectivity of the Mandelbrot set; results of Douady-Hubbard on quadratic-like maps; Yoccoz's results on renormalization and puzzles. The density of hyperbolicity in the family of real quadratic polynomials was proved separately by the author of the survey and in the joint paper with J. Graczyk and G. Swiatek. Both papers combine real and complex methods based on a renormalization theory presented in Sections 1.4 and 1.9. A renormalizable map is a unimodal map that has a periodic interval; its combinatorics is the order of intermediate images of the latter interval. The Feigenbaum-Coullet-Tresser renormalization conjecture asserts that for every given combinatorics the renor…
description: scientific article; zbMATH DE number 6094023