Deviation for interval exchange transformations

Consider a long piece of a trajectory x, T(x), T(T(x)), …, Tⁿ⁻¹(x) of an interval exchange transformation T. A generic interval exchange transformation is uniquely ergodic. Hence, the ergodic theorem predicts that the number χᵢ(x,n) of visits of our trajectory to the ith subinterval would be approximately λᵢ n. Here λᵢ 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 ith subinterval Xᵢ from one predicted by the ergodic theorem. We prove that for almost all interval exchange transformations the following bound is valid: maxx∈ X 1≤ i≤ m n→ +∞ log | χᵢ(x,n) -λᵢn|/log n = θ₂/θ₁ < 1. Roughly speaking the error term is bounded by nθ₂/θ₁. The numbers 0≤ θ₂ < θ₁ depend only on the permutation π 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(π)/θ₁(π)+ε for all n≥ N(ε), where N(ε) does not depend on the starting point x∈ X.