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### Open Access Journals

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.

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).

DCDS

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

JMD

Suppose $f\colon M\to M$ is a $C^{1+\alpha}$ $(\alpha>0)$ diffeomorphism on a compact smooth orientable manifold $M$ of dimension 2, and let $\mu_\Psi$ be an equilibrium measure for a Hölder-continuous potential $\Psi\colon M\to \mathbb R$. We show that if $\mu_\Psi$ has positive measure-theoretic entropy, then $f$ is measure-theoretically isomorphic mod $\mu_\Psi$ to the product of a Bernoulli scheme and a finite rotation.

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