Multifractal analysis for quotients of Birkhoff sums for countable Markov maps

TL;DR

This paper investigates the multifractal structure of quotients of Birkhoff averages in countable Markov maps, establishing a variational principle, analyzing spectrum regularity, and illustrating applications to specific maps and flows.

Contribution

It introduces a variational principle for Hausdorff dimension of level sets and demonstrates the spectrum's analytic variation under certain conditions, with applications to Manneville-Pomeau maps and continued fractions.

Findings

Hausdorff dimension variational principle established

Spectrum can be discontinuous in non-uniform hyperbolic maps

Results extend to suspension flows and continued fractions

Abstract

This paper is devoted to study multifractal analysis of quotients of Birkhoff averages for countable Markov maps. We prove a variational principle for the Hausdorff dimension of the level sets. Under certain assumptions we are able to show that the spectrum varies analytically in parts of its domain. We apply our results to show that the Birkhoff spectrum for the Manneville-Pomeau map can be discontinuous, showing the remarkable differences with the uniformly hyperbolic setting. We also obtain results describing the Birkhoff spectrum of suspension flows. Examples involving continued fractions are also given.