The Poisson boundary of Teichmüller space
type: Item
instance of: scholarly article
title: The Poisson boundary of Teichmüller space
zbMATH Open document ID: 0953.30029
author: Vadim A. Kaimanovich
author: Howard Masur
published in: Journal of Functional Analysis
publication date: 1998-11-29T00:00:00Z
review text: Let \(T_g\) be the Teichmüller space of conformal structures on a compact Riemann surface \(M\) of genus \(g\geq 2\) and \(\Gamma\) the corresponding mapping class group. In analogy to the case \(g=1\), when \(T_1\) is given by the hyperbolic plane \(H^2\), the authors prove an integral representation Poisson formula for bounded harmonic functions on \(T_g\) in terms of their boundary values on PMF, the space of projective measured foliations on \(M\) which is the boundary of \(T_g\) in the Thurston \(\Gamma\)-compactification of \(T_g\). Based on anterior papers, they construct the Poisson boundary of a Markov operator \(P\) on \(T_g\) and identify it with PMF. Given a family of \(\Gamma\)-invariant probability measures \(\pi_x\), \(x\in T_g\), the authors consider the Markov chain with transition probabilities \(\pi_x\), the Markov operator \(P\), \(Pf(x)= \langle f,\pi_x\rangle\), and the \(P\)-harmonic functions \(f= Pf\) on \(T_g\). The Poisson boundary of \(P\) is given by a measure \(\Gamma\)-space \((\partial P,\nu)\) with a family of probability measures \(\nu_x\sim\nu\) on \(\partial P\), \(x\in T_g\), such that the Poisson formula \(f(x)= \langle\widehat f,\nu_x\rangle\) gives an isometry between the space \(H^\infty(T_g, P)\) of bounded \(P\)-harmonic functions and \(L^\infty(\partial P,\nu)\). Existence and uniqueness of \((\partial P,\nu)\) under assumptions on \(\pi_x\) and \(\nu_x\) have been established by \textit{V. A. Kaimanovich} in 1991 [Isr. J. Math. 89, No. 1-3, 77-134 (1995; Zbl 0843.43001)]. Under additional conditions on \(P\) and if \(\overline P\) (the quotient operator on the moduli space) is Harris recurrent (H.r.), the authors prove that there exists a unique family of \(\Gamma\)-invariant probability measures \(\lambda_x\), \(x\in T_g\), on PMF, stationary, pairwise equivalent, concentrated on the set of the uniquely ergodic foliations \(\text{UE}\subset \text{PMF}\) (Theorem 4.1). If in addition \(\overline P\) is positively H.r., th…
reviewed by: Cabiria Andreian Cazacu
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 60J50
Mathematics Subject Classification ID: 60J45
zbMATH DE Number: 1214155
zbMATH Keywords: Teichmüller space
zbMATH Keywords: Thurston boundary
zbMATH Keywords: Markov operator
zbMATH Keywords: Poisson boundary
zbMATH Keywords: harmonic function
zbMATH Keywords: Harnack inequality
MaRDI profile type: Publication
full work available at URL: jfan.1998.3252
OpenAlex ID: W2148625165
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DOI: 10.1006/JFAN.1998.3252
description: scientific article; zbMATH DE number 1214155