An upper bound for the box dimension of the hyperbolic dynamics via unstable pressure

TL;DR

This paper establishes an upper bound for the box dimension of hyperbolic sets using unstable pressure, providing new insights into the geometric complexity of hyperbolic dynamical systems.

Contribution

It introduces a novel upper bound for the box dimension of hyperbolic sets via sub-additive unstable pressure, and offers a new expression for topological pressure.

Findings

Upper bound for box dimension of hyperbolic sets derived

New expression for topological pressure provided

Method based on sub-additive unstable pressure

Abstract

In this paper, we utilize the sub-additive unstable pressure to give an upper bound for the upper box dimension of the $C^1$ hyperbolic set on unstable manifolds. As a by-product, we give a new expression of the topological pressure. This work is inspired by \cite{FS, WCZ, CliPZ}.