Multifractal analysis of Birkhoff averages for countable Markov maps

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.

27 pages. Substantial changes have been made to Theorem 1.2 and sections 4,5 and 6. Some minor corrections have been made elsewhere

References in corpus (5)

• A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates
• Strong renewal theorems and Lyapunov spectra for $\alpha$-Farey and $\alpha$-Lüroth systems
• Regularity of multifractal spectra of conformal iterated function systems
• On the frequency of partial quotients of regular continued fractions
• The Lyapunov spectrum is not always concave

Cited by in corpus (8)

• The thermodynamic approach to multifractal analysis
• Subexponentially increasing sums of partial quotients in continued fraction expansions
• Large deviations for denominators of continued fractions
• How many inflections are there in the Lyapunov spectrum?
• Thermodynamic formalism for transient dynamics on the real line
• Counting the Lyapunov inflections in piecewise linear systems
• Pressure, Poincaré series and box dimension of the boundary
• Limit theorems for sums of products of consecutive partial quotients of continued fractions