Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents

Abstract

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 kown 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 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> </mml:math> , and the genus of corresponding surface is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> , there are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mi>g</mml:mi> </mml:mrow> </mml:math> Lyapunov exponents, which are equal to zero, while the remaining <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>g</mml:mi> </mml:mrow> </mml:math> ones are distributed into pairs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>θ</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>-</mml:mo> <mml:mi>i</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . We present an explicit formula for the largest one.