Deviation for interval exchange transformations

Abstract. Consider a long piece of trajectory x, T (x), T (T (x)),..., Tn−1(x) of an interval exchange transformation T. A generic interval exchange trans-formation is uniquely ergodic. Hence the ergodic theorem predicts that the number χi(x, n) of visits of our trajectory to the i-th subinterval would be approximately λin. Here λi is the length of 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 Xi from one predicted by the ergodic theorem. We prove that for almost all interval exchange transformations the following bound is valid: max x∈X 1≤i≤m lim sup n→+∞ log |χi(x, n) − λin| logn θ2 θ1 < 1 Roughly speaking the error term is bounded by nθ2/θ1. The numbers 0 ≤ θ2 < θ1 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 θ2 = 0. In the case of exchange of three and more intervals the numbers θ1, θ2 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 ∈ X in the following sense. For any ε> 0 the ratio of logarithms would be less than θ2(pi)/θ1(pi) + ε for all n ≥ N(ε), where N(ε) does not depend on the starting point x ∈ X. 1