Topological pressure of free semigroup actions for non-compact sets and Bowen's equation

TL;DR

This paper extends Bowen's equation to non-compact sets under free semigroup actions, introducing new notions of topological pressure and establishing key relationships with Hausdorff dimension and skew-product transformations.

Contribution

It generalizes Bowen's equation to non-compact sets for free semigroup actions, defining new pressure concepts and exploring their properties and applications.

Findings

Characterizes Hausdorff dimension via topological pressure and Bowen equation.

Estimates topological pressure for free semigroup actions on arbitrary sets.

Establishes relationships between pressures of skew-product transformations and free semigroup actions.

Abstract

Climenhaga showed the applicability of Bowen equation to arbitrary subset of a compact metric space. The main purpose of this paper is to generalize the main result of Climenhaga to free semigroup actions for non-compact sets. We introduce the notions of the topological pressure and lower and upper capacity topological pressure of a free semigroup action for non-compact sets by using the Caratheodory- Pesin structure. Some properties of these notions are given, followed by three main results. One is to characterize the Hausdorff dimension of arbitrary subset in term of the topological pressure by Bowen equation, whose points have the positive lower Lyapunov exponents and satisfy a tempered contraction condition, the other is the estimation of topological pressure of a free semigroup action on arbitrary subset of X and the third is the relationship between the upper capacity topological pressure of a skew-product transformation and the upper capacity topological pressure of a free semigroup action with respect to arbitrary subset.