A temporal central limit theorem for real-valued cocycles over rotations

Abstract

We consider deterministic random walks on the real line driven by irrational rotations, or equivalently, skew product extensions of a rotation by α where the skewing cocycle is a piecewise constant mean zero function with a jump by one at a point $β$. When $α$ is badly approximable and $β$ is badly approximable with respect to α, we prove a Temporal Central Limit theorem (in the terminology recently introduced by D. Dolgopyat and O. Sarig), namely we show that for any fixed initial point, the occupancy random variables, suitably rescaled, converge to a Gaussian random variable. This result generalizes and extends a theorem by J. Beck for the special case when $α$ is quadratic irrational, $β$ is rational and the initial point is the origin, recently reproved and then generalized to cover any initial point using geometric renormalization arguments by Avila–Dolgopyat–Duryev–Sarig (Israel J. Math. 207 (2015) 653–717) and Dolgopyat–Sarig (J. Stat. Phys. 166 (2017) 680–713). We also use renormalization, but in order to treat irrational values of $β$, instead of geometric arguments, we use the renormalization associated to the continued fraction algorithm and dynamical Ostrowski expansions. This yields a suitable symbolic coding framework which allows us to reduce the main result to a CLT for non homogeneous Markov chains.