On fast Birkhoff averaging

Abstract

We study the pointwise behavior of Birkhoff sums $S_n\phi(x)$ on subshifts of finite type for Hölder continuous functions $\phi$. In particular, we show that for a given equilibrium state $\mu$ associated to a Hölder continuous potential, there are points $x$ such that $S_n\phi(x) - n \mathbb{E}_\mu \phi \sim a n^\beta$ 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 $\mathbb{E}_\nu \phi = \mathbb{E}_\mu \phi$. 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 $(a, \beta)$, 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.