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On fast Birkhoff averaging

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Author

Summary, in English

We study the pointwise behavior of Birkhoff sums S(n)phi(x) on subshifts of finite type for Holder continuous functions phi. In particular, we show that for a given equilibrium state mu associated to a Holder continuous potential, there are points x such that S(n)phi(x) - nE(mu)phi similar to an(beta) for any a > 0 and 0 < beta < 1. Actually the Hausdorff dimension of the set of such points is bounded from below by the dimension of mu and it is attained by some maximizing equilibrium state nu such that E(nu)phi = E(mu)phi. On such points the ergodic average n(-1) S(n)phi(x) converges more rapidly than predicted by the Birkhoff Theorem, the Law of the Iterated Logarithm and the Central Limit Theorem. All these sets, for different choices (alpha, beta), are distinct but have the same dimension. This reveals a rich multifractal structure of the symbolic dynamics. As a consequence, we prove that the set of uniform recurrent points, which are close to periodic points, has full dimension. Applications are also given to the study of syndetic numbers, Hardy-Weierstrass functions and lacunary Taylor series.

Department/s

Publishing year

2003

Language

English

Pages

443-467

Publication/Series

Mathematical Proceedings of the Cambridge Philosophical Society

Volume

135

Issue

3

Document type

Journal article

Publisher

Cambridge University Press

Topic

  • Mathematics

Status

Published

Research group

  • Dynamical systems

ISBN/ISSN/Other

  • ISSN: 1469-8064