Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets

TL;DR

This paper develops a thermodynamic formalism for countable symbolic spaces and applies it to analyze the multifractal structure and dimension theory of typical infinitely generated self-affine sets, extending Falconer's formula.

Contribution

It introduces a generalized thermodynamic formalism for quasi-multiplicative potentials and derives new multifractal dimension formulas for Birkhoff averages on infinitely generated self-affine sets.

Findings

Generalization of Falconer's dimension formula to infinitely generated sets

Formula for Hausdorff dimension of Birkhoff average level sets

Existence of ergodic measures of full dimension under certain conditions

Abstract

We develop a thermodynamic formalism for quasi-multiplicative potentials on a countable symbolic space and apply these results to the dimension theory of infinitely generated self-affine sets. The first application is a generalisation of Falconer's dimension formula to include typical infinitely generated self-affine sets and show the existence of an ergodic invariant measure of full dimension whenever the pressure function has a root. Considering the multifractal analysis of Birkhoff averages of general potentials $\Phi$ taking values in $\R^{\N}$, we give a formula for the Hausdorff dimension of $J_\Phi(\alpha)$, the $\alpha$-level set of the Birkhoff average, on a typical infinitely generated self-affine set. We also show that for bounded potentials $\Phi$, the Hausdorff dimension of $J_\Phi(\alpha)$ is given by the maximum of the critical value for the pressure and the supremum of Lyapunov dimensions of invariant measures $\mu$ for which $\int\Phi\,d\mu=\alpha$. Our multifractal results are new in both the finitely generated and the infinitely generated setting.