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

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.