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A variation formula for the topological entropy of convex-cocompact manifolds

Published online by Cambridge University Press:  25 November 2010

SAMUEL TAPIE
SAMUEL TAPIE*
Affiliation:
Laboratoire Jean Leray, Université de Nantes, 2 rue de la Houssinière - BP 92208, F-44322 Nantes Cedex 3, France (email: [email protected])
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Abstract

Let (M,gλ) be a 𝒞2-family of complete convex-cocompact metrics with pinched negative sectional curvatures on a fixed manifold. We show that the topological entropy htop(gλ) of the geodesic flow is a 𝒞1 function of λ and we give an explicit formula for its derivative. We apply this to show that if ρλ(Γ)⊂PSL2(ℂ) is an analytic family of convex-cocompact faithful representations of a Kleinian group Γ, then the Hausdorff dimension of the limit set Λρλ(Γ) is a 𝒞1 function of λ. Finally, we give a variation formula for Λρλ (Γ).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 6 , December 2011 , pp. 1849 - 1864
Copyright
Copyright © Cambridge University Press 2011

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