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

TL;DR

This paper introduces a new concept of topological pressure for semigroup actions on non-compact sets using Carathéodory-Pesin structures, and applies Bowen's equation to determine the Hausdorff dimension of specific subsets with positive Lyapunov exponents.

Contribution

It develops a novel framework for topological pressure in semigroup dynamics and links it to Hausdorff dimension via Bowen's equation for certain non-compact sets.

Findings

Defines topological pressure for semigroup actions on arbitrary sets.

Characterizes Hausdorff dimension using Bowen's equation for subsets with positive Lyapunov exponents.

Establishes a connection between pressure, dimension, and dynamical properties in non-compact settings.

Abstract

Inspired to the work of Ma and Wu\cite{Ma} and Climenhaga\cite{Climenhaga}, we introduce the new nation of topological pressure of a semigroup of maps by using the Carath\'{e}odory-Pesin structure (C-P structure) with respect to arbitrary subset in this paper. Moreover, by Bowen's equation, we characterize the Hausdorff dimension of an arbitrary subset, where the points of the subset have the positive lower Lyapunov exponents and satisfy a so called tempered contraction condition.