×

zbMATH — the first resource for mathematics

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.

MSC:
28D05 Measure-preserving transformations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] P. ARNOUX, G. LEVITT, Sur l’unique ergodicité des 1-formes fermées singulières, Inventiones Math., 85 (1986), 141-156 & 645-664. · Zbl 0577.58021
[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. · Zbl 0801.11036
[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 0488.70015
[4] A.B. KATOK, Invariant measures of flows on oriented surfaces, Soviet Math. Dokl., 14 (1973), 1104-1108. · Zbl 0298.28013
[5] M. KEANE, Interval exchange transformations, Math. Z., 141, (1975), 25-31. · Zbl 0278.28010
[6] S.P. KERCKHOFF, Simplicial systems for interval exchange maps and measured foliations, Ergod. Th. & Dynam. Sys., 5 (1985), 257-271. · Zbl 0597.58024
[7] S. KERCKHOFF, H. MASUR, J. SMILLIE, Ergodicity of billiard flows and quadratic differentials, Annals of Math., 124 (1986), 293-311. · Zbl 0637.58010
[8] A. MAIER, On trajectories on closed orientable surfaces, Mat. Sbornik, 12 (1943), 71-84. · Zbl 0063.03856
[9] H. MASUR, Interval exchange transformations and measured foliations, Annals of Math., 115-1 (1982), 169-200. · Zbl 0497.28012
[10] A. NOGUEIRA, D. RUDOLPH, Topological weakly mixing of interval exchange maps, to appear. · Zbl 0958.37010
[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 0236.93034
[13] G. RAUZY, Echanges d’intervalles et transformations induites, Acta Arith., 34 (1979), 315-328. · Zbl 0414.28018
[14] S. SCHWARTZMAN, Asymptotic cycles, Annals of Mathematics, 66 (1957), 270-284. · Zbl 0207.22603
[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. · Zbl 0486.28014
[17] W.A. VEECH, The metric theory of interval exchange transformations I. Generic spectral properties, Amer. Journal of Math., 106 (1984), 1331-1359. · Zbl 0631.28006
[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. · Zbl 0631.28007
[19] W.A. VEECH, The Teichmüller geodesic flow, Annals of Mathematics, 124 (1986), 441-530. · Zbl 0658.32016
[20] W.A. VEECH, Moduli spaces of quadratic differentials, Journal d’Analyse Mathématique, 55 (1990), 117-171. · Zbl 0722.30032
[21] M. WOJTKOWSKI, Invariant families of cones and Lyapunov exponents, Ergod. Th. & Dynam. Sys., 5 (1985), 145-161. · Zbl 0578.58033
[22] A. ZORICH, Asymptotic flag of an orientable measured foliation on a surface, in “Geometric Study of Foliations”, World Sci., 1994, 479-498.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.