## Journals

- Advances in Mathematics of Communications
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DCDS

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.

JMD

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.

DCDS

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).

JMD

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