Unique ergodicity for non-uniquely ergodic horocycle flows
François Ledrappier Omri Sarig
Discrete & Continuous Dynamical Systems - A 2006, 16(2): 411-433 doi: 10.3934/dcds.2006.16.411
We consider the horocycle flow associated to a $\Z^d$-cover of a compact hyperbolic surface. Such flows have no finite invariant measures, and infinitely many infinite ergodic invariant Radon measures. We prove that, up to normalization, only one of these infinite measures admits a generalized law of large numbers, and we identify such laws.
keywords: rational ergodicity ergodic theorems. horocycle flows geometrically infinite
Volume entropy of hyperbolic buildings
François Ledrappier Seonhee Lim
Journal of Modern Dynamics 2010, 4(1): 139-165 doi: 10.3934/jmd.2010.4.139
We characterize the volume entropy of a regular building as the topological pressure of the geodesic flow on an apartment. We show that the entropy maximizing measure is not Liouville measure for any regular hyperbolic building. As a consequence, we obtain a strict lower bound on the volume entropy in terms of the branching numbers and the volume of the boundary polyhedrons.
keywords: building topological entropy geodesic flow. volume entropy volume growth
Fluctuations of ergodic sums for horocycle flows on $\Z^d$--covers of finite volume surfaces
François Ledrappier Omri Sarig
Discrete & Continuous Dynamical Systems - A 2008, 22(1&2): 247-325 doi: 10.3934/dcds.2008.22.247
We study the almost sure asymptotic behavior of the ergodic sums of $L^1$--functions, for the infinite measure preserving system given by the horocycle flow on the unit tangent bundle of a $\Z^d$--cover of a hyperbolic surface of finite area, equipped with the volume measure. We prove rational ergodicity, identify the return sequence, and describe the fluctuations of the ergodic sums normalized by the return sequence. One application is a 'second order ergodic theorem': almost sure convergence of properly normalized ergodic sums, subject to a certain summability method (the ordinary pointwise ergodic theorem fails for infinite measure preserving systems).
keywords: ergodic theorems. horocycle flow geometrically infinite
On Omri Sarig's work on the dynamics on surfaces
François Ledrappier
Journal of Modern Dynamics 2014, 8(1): 15-24 doi: 10.3934/jmd.2014.8.15
keywords: Brin prize Sarig Markov partitions surface diffeomorphisms horocycle flow.

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