Bowen's equation in the non-uniform setting

TL;DR

This paper extends Bowen's equation to more general settings, allowing the calculation of Hausdorff dimensions for sets with positive Lyapunov exponents under weaker conditions, including maps with parabolic points.

Contribution

It generalizes Bowen's equation to arbitrary subsets with positive Lyapunov exponents, broadening its applicability beyond conformal expanding maps.

Findings

Applicable to sets with positive Lyapunov exponents

Allows dimension computation for maps with parabolic points

Relates Hausdorff dimension to topological entropy in symbolic spaces

Abstract

We show that Bowen's equation, which characterises the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In particular, we consider an arbitrary subset Z of a compact metric space and require only that the lower Lyapunov exponents be positive on Z, together with a tempered contraction condition. Among other things, this allows us to compute the dimension spectrum for Lyapunov exponents for maps with parabolic periodic points, and to relate the Hausdorff dimension to the topological entropy for arbitrary subsets of symbolic space with the appropriate metric.