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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: [email protected]
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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
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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