Francois Ledrappier (University of Notre Dame)

Distributions of horocycles on abelian covers.

Abstract. We consider the horocycle flow $h_s$ on an abelian $d$ -dimensional cover of a compact hyperbolic surface. The invariant ergodic Radon measures form a continuous family indexed by ${\mathbb{R}}^d$ . We show that out of them, only the Lebesgue measure $m_0$ is rationally ergodic. Moreover, the Lebesgue measure satisfies the following ergodic theorem: for every $f\in L^1$ , $m_0$ -almost every $\omega$ :

$$\lim _{N \to \infty } \frac {1}{\ln \ln N} \int _3^\infty \frac {... ...int _0^T f(h_s\omega ) ds \right) \frac {dT}{T\ln T}\quad = \quad \int f dm_0,$$

where $a(T) = (\ln T)^{d/2} /T$ and $C$ is an explicit constant. This is a joint work with O . Sarig.