Transfer operator and conformal measures for a class of maps having covering property

TL;DR

This paper extends Walters' theory to analyze conformal measures and Hausdorff dimension for a class of maps with a covering property, using the Perron-Frobenius-Ruelle operator in a metric space setting.

Contribution

It develops a framework for conformal measures and Bowen's formula for maps with covering properties, expanding the scope of thermodynamic formalism.

Findings

Existence of conformal measures for the class of maps studied.

Establishment of Bowen's formula relating Hausdorff dimension and Poincaré exponent.

Application of Perron-Frobenius-Ruelle operator in a metric space context.

Abstract

Let $(X,d)$ be a metric space and $X_0$ be an open and dense subset of $X$. We develop the Walters' theory and discuss the existence of conformal measures in terms of the Perron-Frobenius-Ruelle operator for a continuous map $T:X_0\rightarrow X$ and the Bowen formula about Hausdorff dimension and Poincar\'e exponent of some invariant subsests for $T$ with some expanding property.