Targets, local weak $\sigma$-Gibbs measures and a generalized Bowen dimension formula

TL;DR

This paper develops a generalized Bowen dimension formula for sets of points in dynamical systems that approximate any point at specified rates, using local weak Gibbs measures on topological Markov chains and applying to various maps including Gauss and Luroth.

Contribution

It introduces a generalized Bowen formula for Carathéodory and Hausdorff dimensions in systems with local weak Gibbs measures, extending results to non-Hölder potentials and intermittent maps.

Findings

Derived Bowen formula for Carathéodory dimension of approximation sets.

Established Bowen-type formula for Hausdorff dimension in Markov transformations.

Applied results to Gauss, Luroth maps, and intermittent systems like Manneville-Pomeau.

Abstract

For a dynamical system, we study the set of points $\cal W$ whose orbit approximates any chosen point at certain specified rates. Our basic setting is that of left shift acting on topological Markov chains endowed with a local weak Gibbs measure. Our rates of recurrence are so fast that the corresponding set $\cal W$ has measure zero, but we obtain a generalized Bowen formula for Carath\'eodory dimension. For the case of Markov transformations with countable partition and big image (BI) property a Bowen-type formula is obtained for the Hausdorff dimension of those exceptional sets. In particular, we apply our general results to Gauss and Luroth maps, a not Bernoulli modification of Gauss map and some inner functions. Since we only require the existence of weak Gibbs measures we can deal with non H\"older potentials and we can also consider intermittent systems as the Manneville-Pomeau map.