Cantor sets constructed by infinitely renormalizable diffeomorphisms
type: Item
instance of: scholarly article
title: Cantor sets constructed by infinitely renormalizable diffeomorphisms
zbMATH Open document ID: 0864.58041
DOI: 10.1016/0166-8641(96)00071-5
author: Naoya Sumi
published in: Topology and its Applications
publication date: 1996-11-07T00:00:00Z
review text: The author investigates the notion of infinitely renormalizable diffeomorphisms in the context of two-dimensional dynamics. Constructions of such diffeomorphisms have been previously discussed by Bowen, Franks and Young. This paper establishes that the regularity of these diffeomorphisms is closely related to the Hausdorff dimension of their attracting, invariant Cantor sets. Indeed, if \(K\) is such a Cantor set and \(f: D\to D\) is an infinitely renormalizable diffeomorphism of the unit disk in \(\mathbb{R}^2\) with \(f\in C^{1+\varepsilon}\), then \(\varepsilon\leq \dim_H(K)\) where \(\dim_H\) denotes the Hausdorff dimension. Moreover, the author shows that if \(f\in C^{2+\varepsilon}\), then \(1+\varepsilon\leq \dim_H(K)\). Furthermore, there are no infinitely renormalizable diffeomorphisms of class \(C^3\).
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 54C70
zbMATH DE Number: 943250
zbMATH Keywords: topological entropy
zbMATH Keywords: minimal set
zbMATH Keywords: infinitely renormalizable diffeomorphisms
zbMATH Keywords: Hausdorff dimension
reviewed by: William J. Satzer Jr.
MaRDI profile type: Publication
full work available at URL: 0166-8641(96)00071-5
OpenAlex ID: W2027257693
cites work: The periodic points of maps of the disk and the interval
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cites work: Henon-like maps with strange attractors: there exist C∞Kupka-Smale diffeomorphisms on S2with neither sinks nor sources
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description: scientific article; zbMATH DE number 943250