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Multifractal analysis of Birkhoff averages for countable Markov maps

Published online by Cambridge University Press:  25 August 2015

GODOFREDO IOMMI  and
THOMAS JORDAN
GODOFREDO IOMMI
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile (PUC), Avenida Vicuña Mackenna 4860, Santiago, Chile email giommi@mat.puc.cl
THOMAS JORDAN
Affiliation:
The School of Mathematics, The University of Bristol, University Walk, Clifton, Bristol BS8 1TW, UK email Thomas.Jordan@bristol.ac.uk
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 8 , December 2015 , pp. 2559 - 2586
Copyright
© Cambridge University Press, 2015 

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