Multifractal analysis of Birkhoff averages for countable Markov maps

TL;DR

This paper develops a multifractal formalism for Birkhoff averages in countable Markov maps, showing the spectrum's real analyticity under certain conditions and applying results to number theory, including Hausdorff dimension calculations.

Contribution

It introduces a multifractal formalism for Birkhoff averages in countable Markov maps and proves the spectrum's real analyticity under regularity assumptions.

Findings

Birkhoff spectrum is real analytic under certain conditions

Hausdorff dimension of points with infinite Birkhoff average computed

Applications to number theory demonstrated

Abstract

In this paper we prove a multifractal formalism of Birkhoff averages for interval maps with countably many branches. Furthermore, we prove that under certain regularity assumptions on the potential the Birkhoff spectrum is real analytic. Applications of these results to number theory are also given. Finally, we compute the Hausdorff dimension of the set of points for which the Birkhoff average is infinite.