Abstract
In the moduli space \({{\mathcal {H}}_g}\) of normalized translation surfaces of genus g, consider, for a small parameter ρ > 0, those translation surfaces which have two non-parallel saddle-connections of length ⩽ ρ. We prove that this subset of \({{\mathcal {H}}_g}\) has measure o(ρ 2) w.r.t. any probability measure on \({{\mathcal {H}}_g}\) which is invariant under the natural action of \({SL(2,\mathbb{R})}\) . This implies that any such probability measure is regular, a property which is important in relation with the recent fundamental work of Eskin–Kontsevich–Zorich on the Lyapunov exponents of the KZ-cocycle.
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References
Avila A., Viana M.: Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture. Acta Math., 198, 1–56 (2007)
Bouw I., Möller M.: Teichmüller curves, triangle groups, and Lyapunov exponents. Annals of Mathematics 172, 139–185 (2010)
V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model, preprint, 2011, arXiv:1107.1810, to appear in Ann. Sci. Éc. Norm. Supér
Eskin A., Kontsevich M., Zorich A.: Lyapunov spectrum of square-tiled cyclic covers. Journal of Modern Dynamics 5, 319–353 (2011)
A. Eskin, M. Kontsevich and A. Zorich. Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, preprint, 2011, arXiv:1112.5872
Eskin A., Masur H.: Asymptotic formulas on flat surfaces. Ergodic Theory and Dynamical Systems 21, 443–478 (2001)
A. Eskin, H. Masur and A. Zorich. Moduli spaces of Abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants, Publications Mathématiques de l’IHÉS, v. 97 (2003), 61–179
Forni G.: Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Annals of Mathematics, 155(1), 1–103 (2002)
Forni G.: A geometric criterion for the non-uniform hyperbolicity of the Kontsevich–Zorich cocycle.. Journal of Modern Dynamics, 5, 355–395 (2011)
J. Grivaux and P. Hubert. Les exposants de Liapounoff du flot de Teichmüller (d’après Eskin-Kontsevich-Zorich), Séminaire Bourbaki 2012–2013, n. 1060.
M. Kontsevich. Lyapunov exponents and Hodge theory. In: The Mathematical Beauty Of Physics (Saclay, 1996), Advanced Series in Mathematical Physics, Vol 24. World Scientific, River Edge (1997) pp. 318–332
Masur H.: Interval exchange transformations and measured foliations. Annals of Mathematics 115, 169–200 (1982)
Masur H.: The growth rate of trajectories of a quadratic differential. Ergodic Theory and Dynamical Systems 10, 151–176 (1990)
Masur H., Smillie J.: Hausdorff dimension of sets of nonergodic foliations. Annals of Mathematics, 134, 445–543 (1991)
C. Matheus, M. Möller and J.-C. Yoccoz, A criterion for the simplicity of the Lyapunov exponents of square-tiled surfaces, preprint (2013), arXiv:1305.2033
D.-M. Nguyen. Volumes of the sets of translation surfaces with small saddle connections, preprint, (2012), arXiv:1211.7314
V. Rokhlin. On the fundamental ideas of measure theory, Mat. Sbornik N.S. 25(67), (1949), 107–150
Veech W.: Gauss measures for transformations on the space of interval exchange maps. Annals of Mathematics 115, 201–242 (1982)
Veech W.: Siegel measures. Annals of Mathematics 148, 895–944 (1998)
J.-C. Yoccoz. Interval exchange maps and translation surfaces, Clay Math. Inst. Summer School on Homogenous Flows, Moduli Spaces and Arithmetic, Pisa (2007), available at http://www.college-de-france.fr/media/equ_dif/UPL15305_PisaLecturesJCY2007.pdf
L.-S. Young. Ergodic theory of differentiable dynamical systems, Real and complex dynamical systems, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Vol. 464, Kluwer Academic Publishers, Dordrecht (1995), pp. 293–336
A. Zorich. Asymptotic flag of an orientable measured foliation on a surface. In: Geometric Study of Foliations. World Scientific, River Edge (1994), pp. 479–498
Zorich A.: Deviation for interval exchange transformations. Ergodic Theory and Dynamical Systems 17, 1477–1499 (1997)
A. Zorich. Flat Surfaces. In: Frontiers in Number Theory, Physics, and Geometry, Springer, Berlin (2006), pp. 437–583
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Avila, A., Matheus, C. & Yoccoz, JC. \({SL(2, \mathbb{R})}\) -invariant probability measures on the moduli spaces of translation surfaces are regular. Geom. Funct. Anal. 23, 1705–1729 (2013). https://doi.org/10.1007/s00039-013-0244-5
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DOI: https://doi.org/10.1007/s00039-013-0244-5