Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents
Annales de l'Institut Fourier, Tome 46 (1996) no. 2, pp. 325-370.

Nous construisons une application sur l’espace des échanges d’intervalles qui généralise l’application classique d’intervalle associée au développement en fraction continue. Cette application est fondée sur l’induction de Rauzy, mais à la différence des fonctions similaires connues jusqu’à présent cette application est ergodique par rapport à une mesure finie absolument continue sur l’espace des échanges d’intervalles. Nous présentons la procédure de calcul de cette mesure fondée sur la technique élaborée par W. Veech pour l’induction de Rauzy.

Nous étudions les exposants de Lyapunov définis par cette application. Soit m le nombre d’intervalles, et soit g le genre de la surface correspondante. Nous montrons qu’il y a m-2g exposants de Lyapunov qui sont égaux à zéro, alors que les autres 2g exposants sont distribués en tant que θ i =-θ m-i+1 . Nous donnons une formule explicite pour l’exposant le plus grand.

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 m, and the genus of corresponding surface is g, there are m-2g Lyapunov exponents, which are equal to zero, while the remaining 2g ones are distributed into pairs θ i =-θ m-i+1 . We present an explicit formula for the largest one.

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     title = {Finite {Gauss} measure on the space of interval exchange transformations. {Lyapunov} exponents},
     journal = {Annales de l'Institut Fourier},
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Zorich, Anton. Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Annales de l'Institut Fourier, Tome 46 (1996) no. 2, pp. 325-370. doi : 10.5802/aif.1517. http://archive.numdam.org/articles/10.5802/aif.1517/

[1] P. Arnoux, G. Levitt, Sur l'unique ergodicité des 1-formes fermées singulières, Inventiones Math., 85 (1986), 141-156 & 645-664. | MR | Zbl

[2] P. Arnoux, A. Nogueira, Mesures de Gauss pour des algorithmes de fractions continues multidimensionnelles, Ann. scient. Éc. Norm. Sup., 4e série, 26 (1993), 645-664. | Numdam | MR | Zbl

[3] G. Benettin, I. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems ; a method for computing all of them. Part 1 : theory. Meccanica (1980), 9-20. | Zbl

[4] A.B. Katok, Invariant measures of flows on oriented surfaces, Soviet Math. Dokl., 14 (1973), 1104-1108. | Zbl

[5] M. Keane, Interval exchange transformations, Math. Z., 141, (1975), 25-31. | MR | Zbl

[6] S.P. Kerckhoff, Simplicial systems for interval exchange maps and measured foliations, Ergod. Th. & Dynam. Sys., 5 (1985), 257-271. | MR | Zbl

[7] S. Kerckhoff, H. Masur, J. Smillie, Ergodicity of billiard flows and quadratic differentials, Annals of Math., 124 (1986), 293-311. | MR | Zbl

[8] A. Maier, On trajectories on closed orientable surfaces, Mat. Sbornik, 12 (1943), 71-84. | Zbl

[9] H. Masur, Interval exchange transformations and measured foliations, Annals of Math., 115-1 (1982), 169-200. | MR | Zbl

[10] A. Nogueira, D. Rudolph, Topological weakly mixing of interval exchange maps, to appear. | Zbl

[11] A. Nogueira, The 3-dimensional Poincaré continued fraction algorithm, preprint ENSL, 93 (1993), 1-25.

[12] V.I. Oseledets, A Multiplicative Ergodic Theorem. Ljapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231. | Zbl

[13] G. Rauzy, Echanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328. | MR | Zbl

[14] S. Schwartzman, Asymptotic cycles, Annals of Mathematics, 66 (1957), 270-284. | MR | Zbl

[15] W.A. Veech, Projective swiss cheeses and uniquely ergodic interval exchange transformations, Ergodic Theory and Dynamical Systems, Vol. I, in Progress in Mathematics, Birkhauser, Boston, 1981, 113-193.

[16] W.A. Veech, Gauss measures for transformations on the space of interval exchange maps, Annals of Mathematics, 115 (1982), 201-242. | MR | Zbl

[17] W.A. Veech, The metric theory of interval exchange transformations I. Generic spectral properties, Amer. Journal of Math., 106 (1984), 1331-1359. | MR | Zbl

[18] W.A. Veech, The metric theory of interval exchange transformations II. Approximation by primitive interval exchanges, Amer. Journal of Math., 106 (1984), 1361-1387. | MR | Zbl

[19] W.A. Veech, The Teichmüller geodesic flow, Annals of Mathematics, 124 (1986), 441-530. | MR | Zbl

[20] W.A. Veech, Moduli spaces of quadratic differentials, Journal d'Analyse Mathématique, 55 (1990), 117-171. | MR | Zbl

[21] M. Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergod. Th. & Dynam. Sys., 5 (1985), 145-161. | MR | Zbl

[22] A. Zorich, Asymptotic flag of an orientable measured foliation on a surface, in “Geometric Study of Foliations”, World Sci., 1994, 479-498.

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