Mixed multifractal spectra of Birkhoff averages for non-uniformly expanding one-dimensional Markov maps with countably many branches

TL;DR

This paper develops formulas for mixed multifractal spectra of Birkhoff averages in non-uniformly expanding Markov maps with countably many branches, revealing new fractal properties of continued fractions and Fuchsian groups.

Contribution

It introduces conditional variational formulas for multifractal spectra in complex dynamical systems with countably many branches, extending previous understanding.

Findings

Derived new fractal-geometric results for continued fractions

Established formulas for multi-cusp winding spectra

Answered a question posed by Pollicott

Abstract

For a Markov map of an interval or the circle with countably many branches and finitely many neutral periodic points, we establish conditional variational formulas for the mixed multifractal spectra of Birkhoff averages of countably many observables, in terms of the Hausdorff dimension of invariant probability measures. Using our results, we are able to exhibit new fractal-geometric results for backward continued fraction expansions of real numbers, answering in particular a question of Pollicott. Moreover, we establish formulas for multi-cusp winding spectra for the Bowen-Series maps associated with finitely generated free Fuchsian groups with parabolic elements.