DCDS
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
DCDS
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
DCDS
Tail-invariant measures for some suspension semiflows
Jon Aaronson Omri Sarig Rita Solomyak
Discrete & Continuous Dynamical Systems - A 2002, 8(3): 725-735 doi: 10.3934/dcds.2002.8.725
We consider suspension semiflows over abelian extensions of one-sided mixing subshifts of finite type. Although these are not uniquely ergodic, we identify (in the "ergodic" case) all tail-invariant, locally finite measures which are quasiinvariant for the semiflow.
keywords: skew-products unique ergodicity horocycle flow. non-arithmeticity tail-invariance Equivalence relations semi-flows infinite measures aperiodicity

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