A compactification of the space of expanding maps on the circle
type: Item
instance of: scholarly article
title: A compactification of the space of expanding maps on the circle
zbMATH Open document ID: 1181.37065
author: Curtis T. McMullen
published in: Geometric and Functional Analysis. GAFA
publication date: 2009-09-14T00:00:00Z
full work available at URL: urn-3:HUL.InstRepos:3426329
review text: Let \(\mathrm{Exp}_d(S^1)\) denote the space of topological covering maps \(f:S^1\rightarrow S^1\) of degree \(d>1\) that preserve Lebesgue measure. In this paper a compactification \(\widehat{\mathrm{Exp}_d}(S^1)\) for the space of expanding maps is introduced. It is well known that every \(f \in \mathrm{Exp}_d(S^1)\) is topologically conjugate to the model mapping \(p_d(t)=d\cdot t \mathrm{mod}~1\). There are \((d-1)\) choices for conjugacy \(\phi\); the choice of one is a marking of \(f\). Using the marking, one can transport Lebesgue measure to an invariant measure \(\nu_{(f,\phi)}=\phi_{\ast}(\lambda)\) for the model mapping \(p_d.\) The author prove that the map \[ \nu : T\widehat{\mathrm{Exp}_d}(S^1)\rightarrow M_d(S^1) \] gives a bijective homeomorphism between the compactified Teichmüller space of expanding maps and the space of invariant probability measures for \(t\mapsto d\cdot t \mathrm{mod}~1.\) The analogous result holds for the space \(\overline{\mathcal{B}}_d\) of expanding Blaschke products on \(S^1.\) Several interesting properties of \(\overline{\mathcal{B}}_d\) are obtained.
Mathematics Subject Classification ID: 37F30
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 37E10
zbMATH DE Number: 5603850
zbMATH Keywords: expanding maps
zbMATH Keywords: Blaschke products
zbMATH Keywords: invariant measures
zbMATH Keywords: complex dynamics
zbMATH Keywords: Teichmüller theory
zbMATH Keywords: harmonic measure
reviewed by: A. Klíč
MaRDI profile type: Publication
OpenAlex ID: W2127403721
DOI: 10.1007/S00039-009-0709-8
description: scientific article; zbMATH DE number 5603850