Deviation for interval exchange transformations

Consider a long piece of a trajectory $x, T(x), T(T(x)), \ldots, T^{n-1}(x)$ of an interval exchange transformation $T$. A generic interval exchange transformation is uniquely ergodic. Hence, the ergodic theorem predicts that the number $\chi_i(x,n)$ of visits of our trajectory to the $i$th subinterval would be approximately $\lambda_i n$. Here $\lambda_i$ is the length of the corresponding subinterval of our unit interval $X$. In this paper we give an estimate for the deviation of the actual number of visits to the $i$th subinterval $X_i$ from one predicted by the ergodic theorem.We prove that for almost all interval exchange transformations the following bound is valid: $$ \max_{\ssty x\in X \atop \ssty 1\le i\le m} \limsup_{n\to +\infty} \frac {\log | \chi_i(x,n) -\lambda_in|}{\log n} = \frac{\theta_2}{\theta_1} < 1. $$ Roughly speaking the error term is bounded by $n^{\theta_2/\theta_1}$. The numbers $0\le \theta_2 < \theta_1$ depend only on the permutation $\pi$ corresponding to the interval exchange transformation (actually, only on the Rauzy class of the permutation). In the case of interval exchange of two intervals we obviously have $\theta_2=0$. In the case of exchange of three and more intervals the numbers $\theta_1, \theta_2$ are the two top Lyapunov exponents related to the corresponding generalized Gauss map on the space of interval exchange transformations.The limit above ‘converges to the bound’ uniformly for all $x\in X$ in the following sense. For any $\varepsilon >0$ the ratio of logarithms would be less than $\theta_2(\pi)/\theta_1(\pi)+\varepsilon $ for all $n\ge N(\varepsilon)$, where $N(\varepsilon)$ does not depend on the starting point $x\in X$.