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

Annales de l'institut Fourier (1996)

- Volume: 46, Issue: 2, page 325-370
- ISSN: 0373-0956

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topZorich, Anton. "Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents." Annales de l'institut Fourier 46.2 (1996): 325-370. <http://eudml.org/doc/75181>.

@article{Zorich1996,

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 $m$, 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 $\theta _i=-\theta _\{m-i+1\}$. We present an explicit formula for the largest one.},

author = {Zorich, Anton},

journal = {Annales de l'institut Fourier},

keywords = {interval exchange transformation; Gauss measure; Rauzy induction; Lyapunov exponents; orientable measured foliation},

language = {eng},

number = {2},

pages = {325-370},

publisher = {Association des Annales de l'Institut Fourier},

title = {Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents},

url = {http://eudml.org/doc/75181},

volume = {46},

year = {1996},

}

TY - JOUR

AU - Zorich, Anton

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

JO - Annales de l'institut Fourier

PY - 1996

PB - Association des Annales de l'Institut Fourier

VL - 46

IS - 2

SP - 325

EP - 370

AB - 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 $\theta _i=-\theta _{m-i+1}$. We present an explicit formula for the largest one.

LA - eng

KW - interval exchange transformation; Gauss measure; Rauzy induction; Lyapunov exponents; orientable measured foliation

UR - http://eudml.org/doc/75181

ER -

## References

top- [1] P. ARNOUX, G. LEVITT, Sur l'unique ergodicité des 1-formes fermées singulières, Inventiones Math., 85 (1986), 141-156 & 645-664. Zbl0577.58021MR87g:58004
- [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. Zbl0801.11036MR95h:11076
- [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. Zbl0488.70015
- [4] A.B. KATOK, Invariant measures of flows on oriented surfaces, Soviet Math. Dokl., 14 (1973), 1104-1108. Zbl0298.28013
- [5] M. KEANE, Interval exchange transformations, Math. Z., 141, (1975), 25-31. Zbl0278.28010MR50 #10207
- [6] S.P. KERCKHOFF, Simplicial systems for interval exchange maps and measured foliations, Ergod. Th. & Dynam. Sys., 5 (1985), 257-271. Zbl0597.58024MR87g:58075
- [7] S. KERCKHOFF, H. MASUR, J. SMILLIE, Ergodicity of billiard flows and quadratic differentials, Annals of Math., 124 (1986), 293-311. Zbl0637.58010MR88f:58122
- [8] A. MAIER, On trajectories on closed orientable surfaces, Mat. Sbornik, 12 (1943), 71-84. Zbl0063.03856
- [9] H. MASUR, Interval exchange transformations and measured foliations, Annals of Math., 115-1 (1982), 169-200. Zbl0497.28012MR83e:28012
- [10] A. NOGUEIRA, D. RUDOLPH, Topological weakly mixing of interval exchange maps, to appear. Zbl0958.37010
- [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. Zbl0236.93034
- [13] G. RAUZY, Echanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328. Zbl0414.28018MR82m:10076
- [14] S. SCHWARTZMAN, Asymptotic cycles, Annals of Mathematics, 66 (1957), 270-284. Zbl0207.22603MR19,568i
- [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. Zbl0486.28014MR83g:28036b
- [17] W.A. VEECH, The metric theory of interval exchange transformations I. Generic spectral properties, Amer. Journal of Math., 106 (1984), 1331-1359. Zbl0631.28006MR87j:28024a
- [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. Zbl0631.28007MR87j:28024b
- [19] W.A. VEECH, The Teichmüller geodesic flow, Annals of Mathematics, 124 (1986), 441-530. Zbl0658.32016MR88g:58153
- [20] W.A. VEECH, Moduli spaces of quadratic differentials, Journal d'Analyse Mathématique, 55 (1990), 117-171. Zbl0722.30032MR92e:32014
- [21] M. WOJTKOWSKI, Invariant families of cones and Lyapunov exponents, Ergod. Th. & Dynam. Sys., 5 (1985), 145-161. Zbl0578.58033MR86h:58090
- [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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