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

### Antti Käenmäki

University of Jyväskylä, Finland### Henry W. J. Reeve

University of Bristol, Great Britain

## 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 $\mathbb{R}^{\mathbb{N}}$, 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.

## Cite this article

Antti Käenmäki, Henry W. J. Reeve, Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets. J. Fractal Geom. 1 (2014), no. 1, pp. 83–152

DOI 10.4171/JFG/3