Subexponential decay of correlations
type: Item
instance of: scholarly article
title: Subexponential decay of correlations
zbMATH Open document ID: 1042.37005
published in: Inventiones Mathematicae
publication date: 2003-08-17T00:00:00Z
review text: Let \((X,{\mathcal B},m,T)\) be a dynamical system, and denote by \(\widehat T\) the transfer operator (Perron-Frobenius operator) associated with the dynamical system. The first assertion is the following renewal equation. Proposition 1. Let the dyamical system be conservative and nonsingular. Assume that \(A\in{\mathcal B}\) has finite positive measure. Let \(T_nf=1_A\widehat T^n(f1_A)\) and \(R_nf=1_A\widehat T^n(f1_{[\phi_A=n]})\), where \(\phi_A(x)\) is the first return time to \(A\) for \(x\in A\). Then for any \(| z| <1\) \[ T(z)=(I-R(z))^{-1},\quad R(z)=\sum_{n=1}^\infty a^nR_n,\quad T(z) =\sum_{n=0}^\infty z^nT_n. \] The main result of this article is: Theorem 1. Let \(T_n\) be bounded linear operators on a Banach space \(\mathcal L\) such that \(T(z)=\sum_{n=0}^\infty z^nT_n\) converges in \(\Hom({\mathcal L},{\mathcal L})\) for every \(| z| <1\). Assume the Renewal equation: for every \(| z| <1\), \(T(z)=(I-R(z))^{-1}\) where \(R(z)=\sum_{n\geq1}z^nR_n\), \(R_n\in\Hom({\mathcal L},{\mathcal L})\) and \(\sum\| R_n\| <\infty\). Spectral gap: the spectum of \(R(1)\) consists of an isolated simple eigenvalue at \(1\) and a compact subset in \(| z| <1\). Aperiodicity: the spectral radius of \(R(z)\) is strictly less than one for all \(| z| \leq 1\) and \(z\neq1\). Let \(P\) be the eigenprojection of \(R(1)\) at \(1\). If \(\sum_{k>n}\| R_k\| =O(1/n^\beta)\) for some \(\beta>2\) and \(PR'(1)\neq0\), then for all \(n\) \[ T_n={1\over\mu}P+{1\over\mu^2}\sum_{k=n+1}^\infty P_k+E_n \] where \(\mu\) is given by \(PR'(1)P=\mu P\), \(P_n=\sum_{l>n}PR_lP\), and \(E_n\in\Hom({\mathcal L},{\mathcal L})\) satisfy \(\| E_n\| =O(1/n^{[\beta]})\). Theorem 2 gives lower bounds for the correlation functions \(s\): For a Markov partition \(\alpha\), let \([a_,\ldots,a_{n-1}]\) be the cylinder generated by \(a_1,\ldots,a_{n-1}\in\alpha\). Let \[ v_n(\phi)=\sup\{| \phi(x)-\phi(y)| : x,y\in[a_0,\ldots,a_{n-1}],a_i\in\alpha\}. \] The dynamical system is called \((T,\alpha)\)-summab…
reviewed by: Makoto Mori
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 37C30
zbMATH DE Number: 1965454
zbMATH Keywords: transfer operator
zbMATH Keywords: renewal equation
zbMATH Keywords: measure-preserving irreducible Markov dynamical system
author: Omri Sarig
MaRDI profile type: Publication
full work available at URL: s00222-002-0248-5
OpenAlex ID: W2062503053
DOI: 10.1007/S00222-002-0248-5
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description: scientific article; zbMATH DE number 1965454