Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow

AbstractThe geodesic flow of a foliated Riemannian manifold (M, F) is studied. The invariance of some smooth measure is established under some geometrical conditions on F. The Lyapunov exponents and the entropy of this flow are estimated. As an application, the non-existence of foliations with ‘short’ second fundamental tensors is obtained on compact negatively curved manifolds.

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Lyapunov exponents in Hilbert geometry

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2012.145 ◽

2012 ◽

Vol 34 (2) ◽

pp. 501-533 ◽

Cited By ~ 3

Author(s):

MICKAËL CRAMPON

Keyword(s):

Lyapunov Exponents ◽

Convex Functions ◽

Geodesic Flow ◽

Regularity Property ◽

Hilbert Geometry

AbstractWe study the Lyapunov exponents of the geodesic flow of a Hilbert geometry. We prove that all of the information is contained in the shape of the boundary at the endpoint of the chosen orbit. We have to introduce a regularity property of convex functions to make this link precise. As a consequence, Lyapunov manifolds tangent to the Lyapunov splitting appear very easily. All of this work can be seen as a consequence of convexity and the flatness of Hilbert geometries.

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Lyapunov exponents for a stochastic analogue of the geodesic flow

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1986-0831190-1 ◽

1986 ◽

Vol 295 (1) ◽

pp. 85-85 ◽

Cited By ~ 7

Author(s):

A. P. Carverhill ◽

K. D. Elworthy

Keyword(s):

Lyapunov Exponents ◽

Geodesic Flow

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A large deviations bound for the Teichmüller flow on the moduli space of abelian differentials

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385710000349 ◽

2010 ◽

Vol 31 (4) ◽

pp. 1043-1071 ◽

Cited By ~ 5

Author(s):

VÍTOR ARAÚJO ◽

ALEXANDER I. BUFETOV

Keyword(s):

Dynamical Systems ◽

Large Deviations ◽

Moduli Space ◽

Geodesic Flow ◽

Large Deviation ◽

Suspension Flows ◽

Abelian Differentials ◽

Quantitative Recurrence ◽

Teichmuller Geodesic Flow ◽

Teichmüller Flow

AbstractLarge deviation rates are obtained for suspension flows over symbolic dynamical systems with a countable alphabet. We use a method employed previously by the first author [Large deviations bound for semiflows over a non-uniformly expanding base. Bull. Braz. Math. Soc. (N.S.)38(3) (2007), 335–376], which follows that of Young [Some large deviation results for dynamical systems. Trans. Amer. Math. Soc.318(2) (1990), 525–543]. As a corollary of the main results, we obtain a large deviation bound for the Teichmüller flow on the moduli space of abelian differentials, extending earlier work of Athreya [Quantitative recurrence and large deviations for Teichmuller geodesic flow. Geom. Dedicata119 (2006), 121–140].

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