Pressure and recurrence

Véronique Maume-Deschamps¹ ; Bernard Schmitt² ; Mariusz Urbański³ ; Anna Zdunik⁴

¹ Laboratoire de Topologie B.P. 47 870 21078 Dijon Cedex, France
² Laboratoire de Topologie B.P. 47 870 21078-Dijon Cedex, France
³ Department of Mathematics University of North Texas Denton, TX 76203-1430, U.S.A.
⁴ Institute of Mathematics Warsaw University Banacha 2 02-097 Warszawa, Poland

Fundamenta Mathematicae, Tome 178 (2003) no. 2, pp. 129-141.

Résumé

We deal with a subshift of finite type and an equilibrium state $\mu $ for a Hölder continuous function. Let $\alpha ^n$ be the partition into cylinders of length $n$. We compute (in particular we show the existence of the limit) $\mathop {\rm lim}_{n\to \infty } n^{-1}\mathop {\rm log}\nolimits \sum _{j=0}^{\tau _n(x)}\mu (\alpha ^n(T^j(x)))$, where $\alpha ^n (T^j(x))$ is the element of the partition containing $T^j(x)$ and $\tau _n(x)$ is the return time of the trajectory of $x$ to the cylinder $\alpha ^n(x)$.

Détail

DOI : 10.4064/fm178-2-3

Keywords: subshift finite type equilibrium state lder continuous function alpha partition cylinders length compute particular existence limit mathop lim infty mathop log nolimits sum tau alpha x where alpha element partition containing tau return time trajectory cylinder alpha