Subdiffusive behavior generated by irrational rotations

François Huveneers (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We study asymptotic distributions of the sums $y_n(x) = \sum_{k=0}^{n-1} \psi(x+k\alpha)$ with respect to the Lebesgue measure, where $\alpha \in \mathbf{R} - \mathbf{Q}$ and where $\psi$ is the 1-periodic function of bounded variation such that $\psi (x) = 1$ if $x \in [0,1/2[$ and $\psi (x) = -1$ if $x\in[1/2,1[$. For every $\alpha\in\mathbf{R} - \mathbf{Q}$, we find a sequence $(n_j)_j \subset \mathbf{N}$ such that $y_{n_j}/\sqrt j$ is asymptotically normally distributed. For $n\ge 1$, let $z_n \in (y_m)_{m\le n}$ be such that $|| z_n ||_{\mathrm L^2} = \max_{m\le n} ||y_m||_{\mathrm L^2}$. If $\alpha$ is of constant type, we show that $z_n / || z_n ||_{\mathrm L^2}$ is also asymptotically normally distributed. We give an heuristic link with the theory of expanding maps of the interval.

Domaines: Probabilités [math.PR]

François Huveneers. Subdiffusive behavior generated by irrational rotations. Ergodic Theory and Dynamical Systems, 2009, 29 (4), pp.1217-1233. ⟨10.1017/S0143385708000680⟩. ⟨hal-00701757⟩