## Abstract

Consider a long piece of a trajectory x, T(x), T(T(x)),..., T^{n-1}l(x) of an interval exchange transformation T. A generic interval exchange transformation is uniquely ergodic. Hence, the ergodic theorem predicts that the number X_{1}(x, n) of visits of our trajectory to the ith subinterval would be approximately λ_{i}n. Here λ_{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 ith subinterval X_{i} from one predicted by the ergodic theorem. We prove that for almost all interval exchange transformations the following bound is valid: (equation presented) Roughly speaking the error term is bounded by n^{θ2/θ1}. The numbers 0 ≤ θ_{2} < θ_{1} 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 θ_{2} = 0. In the case of exchange of three and more intervals the numbers θ_{1}, θ_{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 ∈ X in the following sense. For any ε > 0 the ratio of logarithms would be less than θ_{2}(π)/θ_{1}(π) + ε for all n ≥ N(ε), where N(ε) does not depend on the starting point x ∈ X.

Original language | English |
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Pages (from-to) | 1477-1499 |

Number of pages | 23 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 17 |

Issue number | 6 |

DOIs | |

Publication status | Published - Dec 1997 |

Externally published | Yes |