The circle and the solenoid
type: Item
instance of: scholarly article
title: The circle and the solenoid
zbMATH Open document ID: 1121.37032
author: Alberto Pinto
author: Dennis Sullivan
published in: Discrete and Continuous Dynamical Systems
publication date: 2007-02-05T00:00:00Z
review text: This very interesting paper gives a comprehensive study of scaling and solenoid functions. These functions represent complete invariants of the smooth conjugacy classes of degree \(d\) smooth expanding circle maps (for \(d\geq 2\)). However, the two notions are not the same. Namely, a scaling function does not determine a moduli space for an expanding circle map, whereas a solenoid function does determine a moduli space. The main results of the paper are the following. (i) A characterization of the sequences of asymptotic length ratios which can occur for systems with Hölder continuous derivative. These ratios are precisely those given by a positive Hölder continuous function \(s\) (the solenoid function) on the Cantor set \(C\) of \(d\)-adic integers satisfying a certain matching condition. Moreover, it is shown that there is a one-to-one correspondence between solonoid functions and affine classes of exponentially fast \(d\)-tilings of the reals. (ii) The explicit calculation of the precise maximal possible level of smoothness for a representation of the expanding circle map, up to diffeomorphic conjugacy, in terms of the function \(s\) and a certain cross ratio function \(t\). For example, in the Lipschitz structure on \(C\) determined by \(s\), the maximal smoothness is \(C^{1+\alpha}\) for \(0<\alpha \leq 1\) if and only if \(s\) is \(\alpha\)-Hölder continuous. The maximal smoothness is \(C^{2+\alpha}\) for \(0<\alpha \leq 1\) if and only if \(t\) is \((1+\alpha)\)-Hölder continuous. An interesting connection with Mostow type rigidity is provided by the fact that \(s\) has to be constant if \(s\) is \(\alpha\)-Hölder continuous for \(\alpha >1\).
reviewed by: Bernd O. Stratmann
Mathematics Subject Classification ID: 37E10
Mathematics Subject Classification ID: 37D20
zbMATH DE Number: 5121901
zbMATH Keywords: maps of the circle
zbMATH Keywords: uniformly hyperbolic systems
zbMATH Keywords: scaling functions
zbMATH Keywords: solenoid functions
zbMATH Keywords: expanding
MaRDI profile type: Publication
full work available at URL: dcds.2006.16.463
OpenAlex ID: W2034154786
DOI: 10.3934/DCDS.2006.16.463
description: scientific article; zbMATH DE number 5121901