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

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.