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Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. (English) Zbl 0853.28007
Summary: We construct a map on the space of interval exchange transformations, which generalizes the classical map on the interval, related to continued fraction expansion. This map is based on Rauzy induction, but unlike its relative known up to now, the map is ergodic with respect to some finite absolutely continuous measure on the space of interval exchange transformations. We present the prescription for calculation of this measure based on a technique developed by W. Veech for Rauzy induction. We study Lyapunov exponents related to this map and show that when the number of intervals is \(m\), and the genus of the corresponding surface is \(g\), there are \(m-2g\) Lyapunov exponents, which are equal to zero, while the remaining \(2g\) ones are distributed into pairs \(\theta_i=-\theta_{m-i+1}\). We present an explicit formula for the largest one.

28D05 Measure-preserving transformations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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