A Limit Theorem for Birkoff sums of non-integrable functions over rotations

Details A Limit Theorem for Birkoff sums of non-integrable functions over rotations A Limit Theorem for Birkoff sums of non-integrable functions over rotations

2007-10-05

arxiv.org/abs/0710.1287v2

Mathematics

Dynamical Systems

24 pages, some typos corrected, final version to appear in ``Probabilistic and Geometric Structures in Dynamics'', edited by K

Scientific paper

We consider Birkhoff sums of functions with a singularity of type 1/x over rotations and prove the following limit theorem. Let $S_N= S_N(\alpha,x)$ be the N^th non-renormalized Birkhoff sum, where $x in [0,1)$ is the initial point, $\alpha\in [0,1)$ is the rotation number and $(\alpha, x)$ are uniformly distributed. We prove that $S_N/N$ has a joint limiting distribution in $(\alpha,x)$ as N tends to infinity. As a corollary, we get the existence of a limiting distribution for certain trigonometric sums.

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