Chapter

Measures and flows on Γ\𝔽<sup><i>d</i></sup>

Jacques Franchi and Yves Le Jan

in Hyperbolic Dynamics and Brownian Motion

Published in print August 2012 | ISBN: 9780199654109
Published online January 2013 | e-ISBN: 9780191745676 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199654109.003.0005

Series: Oxford Mathematical Monographs

Measures and flows on Γ\𝔽d

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This chapter considers measures of Γ-invariant sets, and establish a mixing theorem for the action of the geodesic and horocyclic flows on square-integrable Γ-invariant functions. A Poincaré inequality (i.e., the existence of a spectral gap) for the Laplacian acting on Γ-invariant functions, Γ being a generic cofinite and geometrically finite Kleinian group is derived.

Keywords: Γ-invariant measure; ergodicity; mixing property; Poincaré inequality; spectral gap

Chapter.  10152 words.  Illustrated.

Subjects: Mathematical and Statistical Physics

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