Convergence of potentials for a transfert operator. Applications to dynamical systems and Markov chains (Q2722491)

The authors start with a transfer operator of the form \(Q f (x) := \sum_{s} u_{s} (x) f(s x)\) where \((u_{s})_{s\in S}\) is a family of positive functions, and \((x\mapsto sx)_{s\in S}\) is a finite or countable family of contractions on a compact space \(E\). With such an operator, a Markov chain taking values in \(E\) and also the potential operator \(G:=\sum _{n} Q^{n}\) are associated. For such a Markov chain, a central limit theorem is proved, together with an estimate for the rate of convergence for the CLT, using results on the rate for martingales proved by \textit{E. Haeusler} [Ann. Probab. 16, No. 1, 275-299 (1988; Zbl 0639.60030)], extending thus a result from \textit{A. Raugi} [Ann. Inst. Henri Poincaré, Probab. Stat. 28, No. 2, 281-309 (1992; Zbl 0752.60054)]. NEWLINENEWLINENEWLINERegularity properties for the potential operator in connection with those of the transfer operator are also investigated. The various regularity properties considered are illustrated on an example of a Markov chain. Applications to the dynamical systems are included, namely by proving a Borel-Cantelli-type lemma for associated stationary sequences [\textit{D. Y. Kleinbock} and \textit{G. A. Margulis}, Invent. Math. 138, No. 3, 451-494 (1999; Zbl 0934.22016) and \textit{W. Philipp}, Pac. J. Math. 20, 109-127 (1967; Zbl 0144.04201)].NEWLINENEWLINEFor the entire collection see [Zbl 0958.00034].