Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure

TL;DR

This paper investigates the Hausdorff dimension of non-conformal repellers and the continuity of sub-additive topological pressure, providing new bounds and methods for analyzing dynamical systems with singular potentials.

Contribution

It introduces a modified Katok's approximation method to establish continuity of sub-additive topological pressure and derives a sharp lower bound for the Hausdorff dimension of non-conformal repellers.

Findings

Established continuity of sub-additive topological pressure.

Derived a sharp lower bound for the Hausdorff dimension.

Constructed invariant sets with properties close to equilibrium measures.

Abstract

Given a non-conformal repeller $\Lambda$ of a $C^{1+\gamma}$ map, we study the Hausdorff dimension of the repeller and continuity of the sub-additive topological pressure for the sub-additive singular valued potentials. Such a potential always possesses an equilibrium state. We then use a substantially modified version of Katok's approximating argument, to construct a compact invariant set on which the corresponding dynamical quantities (such as Lyapunov exponents and metric entropy) are close to that of the equilibrium measure. This allows us to establish continuity of the sub-additive topological pressure and obtain a sharp lower bound of the Hausdorff dimension of the repeller. The latter is given by the zero of the super-additive topological pressure