On fast Birkhoff averaging

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Abstract

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.

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  • Mathematics
Original languageEnglish
Pages (from-to)443-467
JournalMathematical Proceedings of the Cambridge Philosophical Society
Volume135
Issue number3
Publication statusPublished - 2003
Publication categoryResearch
Peer-reviewedYes