Multifractal analysis of Birkhoff averages for countable Markov maps

Godofredo Iommi; Thomas Jordan

doi:10.1017/etds.2015.44

Multifractal analysis of Birkhoff averages for countable Markov maps

Godofredo Iommi, Thomas Jordan

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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 assumptions the Birkhoff spectrum is real analytic. We also show that new phenomena occur; indeed, the spectrum can be constant or it can have points where it is not analytic. Conditions for these to happen are obtained. 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.

Original language	English
Pages (from-to)	2559-2586
Number of pages	28
Journal	Ergodic Theory and Dynamical Systems
Volume	35
Issue number	8
Early online date	25 Aug 2015
DOIs	https://doi.org/10.1017/etds.2015.44
Publication status	Published - Dec 2015

Keywords

math.DS
37C45

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This is the author accepted manuscript (AAM). The final published version (version of record) is available online via Cambridge University Press at https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/multifractal-analysis-of-birkhoff-averages-for-countable-markov-maps/A67DFA719D190F07D6AE2BD89CC3BDC0 . Please refer to any applicable terms of use of the publisher.
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AB - In this paper we prove a multifractal formalism of Birkhoff averages for interval maps with countably many branches. Furthermore, we prove that under certain assumptions the Birkhoff spectrum is real analytic. We also show that new phenomena occur; indeed, the spectrum can be constant or it can have points where it is not analytic. Conditions for these to happen are obtained. 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.

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JO - Ergodic Theory and Dynamical Systems

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ER -

Multifractal analysis of Birkhoff averages for countable Markov maps

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