Pointwise dimension for a class of measures on limit sets

TL;DR

This paper investigates the pointwise dimension of a new class of projection measures on fractal limit sets, establishing existence, exact dimensionality, and formulas involving Lyapunov exponents and entropy, with applications to self-conformal measures.

Contribution

It introduces a new class of measures on fractals, proves their pointwise dimension exists and is exact, and derives geometric formulas for these dimensions and projection entropy.

Findings

Pointwise dimension exists almost everywhere for the measures.

Measures are proven to be exact dimensional.

Derived geometric formulas for pointwise dimension and projection entropy.

Abstract

We study the pointwise dimension for a new class of projection measures on arbitrary fractal limit sets without separation conditions. We prove that the pointwise dimension exists a.e. for this class of measures associated to equilibrium states, and it is given by a formula in terms of Lyapunov exponents and a certain type of entropy. Thus these measures are exact dimensional. Self-conformal measures belong to the above class of measures, and this allows us to obtain a new geometric formula for their pointwise dimension. Thus for self-conformal measures we obtain also a geometric formula for their projection entropy.