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Typical properties of periodic Teichmüller geodesics: Lyapunov exponents

Part of: Riemann surfaces Smooth dynamical systems: general theory

Published online by Cambridge University Press:  17 November 2021

URSULA HAMENSTÄDT
URSULA HAMENSTÄDT*
Affiliation:
Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
*
e-mail: ursula@math.uni-bonn.de
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Abstract

Consider a component ${\cal Q}$ of a stratum in the moduli space of area-one abelian differentials on a surface of genus g. Call a property ${\cal P}$ for periodic orbits of the Teichmüller flow on ${\cal Q}$ typical if the growth rate of orbits with property ${\cal P}$ is maximal. We show that the following property is typical. Given a continuous integrable cocycle over the Teichmüller flow with values in a vector bundle $V\to {\cal Q}$, the logarithms of the eigenvalues of the matrix defined by the cocycle and the orbit are arbitrarily close to the Lyapunov exponents of the cocycle for the Masur–Veech measure.

Keywords

abelian differentialsTeichmüller flowperiodic orbitsLyapunov exponentsequidistribution

MSC classification

Primary: 37C40: Smooth ergodic theory, invariant measures 37C27: Periodic orbits of vector fields and flows
Secondary: 30F60: Teichmüller theory

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 2 , February 2023 , pp. 556 - 584
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Athreya, J., Bufetov, A., Eskin, A. and Mirzakhani, M.. Lattice point asymptotics and volume growth on Teichmüller space. Duke Math. J. 161 (2012), 1055–1111.Google Scholar
Bowen, R.. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429–460.CrossRefGoogle Scholar
Canary, R., Epstein, D. and Green, P.. Notes on Notes of Thurston (London Mathematical Society Lecture Note Series, 328). Cambridge University Press, Cambridge, 2006, pp. 1–111.Google Scholar
Eskin, A. and Mirzakhani, M.. Counting closed geodesics in moduli space. J. Mod. Dyn. 5 (2011), 71–105.CrossRefGoogle Scholar
Eskin, A., Mirzakhani, M. and Rafi, K.. Counting closed geodesics in strata. Invent. Math. 215 (2019), 535–607.Google Scholar
Hamenstädt, U.. Train tracks and the Gromov boundary of the complex of curves. Spaces of Kleinian Groups (London Mathematical Society Lecture Note Series, 329). Eds. Minsky, Y., Sakuma, M. and Series, C.. Cambridge University Press, Cambridge, 2006, pp. 187–207.CrossRefGoogle Scholar
Hamenstädt, U.. Invariant Radon measures on measured lamination space. Invent. Math. 176 (2009), 223–273.CrossRefGoogle Scholar
Hamenstädt, U.. Stability of quasi-geodesics in Teichmüller space. Geom. Dedicata 146 (2010), 101–116.CrossRefGoogle Scholar
Hamenstädt, U.. Bowen’s construction for the Teichmüller flow. J. Mod. Dyn. 7 (2013), 489–526.Google Scholar
Hamenstädt, U.. Counting periodic orbits in the thin part of strata. Preprint, 2016.Google Scholar
Hubbard, J. and Masur, H.. Quadratic differentials and foliations. Acta Math. 142 (1979), 221–274.CrossRefGoogle Scholar
Klarreich, E.. The boundary at infinity of the curve complex and the relative Teichmüller space. Preprint, 2018, arXiv:1803.10339.Google Scholar
Kontsevich, M. and Zorich, A.. Connected components of the moduli space of Abelian differentials with prescribed singularities. Invent. Math. 153 (2003), 631–678.CrossRefGoogle Scholar
Lanneau, E.. Connected components of the strata of the moduli space of quadratic differentials. Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 1–56.CrossRefGoogle Scholar
Levitt, G.. Foliations and laminations on hyperbolic surfaces. Topology 22 (1983), 119–136.CrossRefGoogle Scholar
Masur, H.. Interval exchange transformations and measured foliations. Ann. Math. 115 (1982), 169–200.CrossRefGoogle Scholar
Margulis, G. A.. On Some Aspects of the Theory of Anosov Systems (Springer Monographs in Mathematics). Springer, Berlin, 2004.Google Scholar
Martelli, B.. An introduction to geometric topology. Preprint, 2016, arXiv:1610.02592.Google Scholar
Masur, H. and Minksy, Y.. Geometry of the complex of curves I: hyperbolicity. Invent. Math. 138 (1999), 103–149.CrossRefGoogle Scholar
Oseledets, V. I.. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
Rafi, K.. Hyperbolicity in Teichmüller space. Geom. Topol. 18 (2014), 391–414.CrossRefGoogle Scholar
Schleimer, S.. Notes on the complex of curves. Work in the public domain, available at s.schleimer@warwick.ac.uk Google Scholar
Veech, W.. The Teichmüller geodesic flow. Ann. Math. 124 (1990), 441–530.CrossRefGoogle Scholar
Wright, A.. From rational billiards to dynamics on moduli space. Bull. Amer. Math. Soc. (N.S) 53 (2016), 41–56.CrossRefGoogle Scholar
Zorich, A.. How do leaves of a closed 1-form wind around a surface?. Pseudoperiodic Topology (American Mathematical Society Translations: Series 2, 197). American Mathematical Society, Providence, RI, 1999, pp. 135–178.Google Scholar