Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains

Free access article

Issue		ESAIM: PS Volume 7, 2003


Page(s)		115 - 146
DOI		10.1051/ps:2003003

ESAIM: P&S, March 2003, Vol. 7, pp. 115-146
DOI: 10.1051/ps:2003003

Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains

Jean-Pierre Conze and Albert Raugi

IRMAR, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France; conze@univ-rennes1.fr.

(Received January 28, 2002.)

Abstract
We present a spectral theory for a class of operators satisfying a weak "Doeblin-Fortet" condition and apply it to a class of transition operators. This gives the convergence of the series $\sum_{k \ge 0} k^r P^k f$ , $r \in \mathbb{N}$ , under some regularity assumptions and implies the central limit theorem with a rate in $n^{- \frac{1}{2} }$ for the corresponding Markov chain. An application to a non uniformly hyperbolic transformation on the interval is also given.

Mathematics Subject Classification. 60J10, 37A05, 37A25

Key words: Transfer operator, convergence of iterates, Markov chains, rate in the TCL for dynamical systems, Borel-Cantelli property, non uniformly hyperbolic map.

© EDP Sciences, SMAI 2003

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