Equidistribution of horocycles on abelian covers of compact hyperbolic surfaces

Abstract:

A point is called generic for a flow preserving an infinite ergodic invariant Radon measure, if its orbit satisfies the conclusion of the ratio ergodic theorem for every pair of continuous functions with compact support and non-zero integrals.

This means, roughly, that the relative time the orbit spends inside a nice set A tends to zero, but that the ratio of the times the orbit spends in two "nice" sets A,B tends to a definite limit, equal to the ratio of the measures of A and B.

The generic points for horocycle flows on hyperbolic surfaces of finite genus are understood, but there are no results in infinite genus. We give such a result, by characterizing the generic points for Z^d--covers. This is joint work with B. Schapira.

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