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Minimality and unique ergodicity for subgroup actions

Shahar Mozes; Barak Weiss — 1998

Annales de l'institut Fourier

Let $G$ be an $ℝ$ -algebraic semisimple group, $H$ an algebraic $ℝ$ -subgroup, and $Γ$ a lattice in $G$ . Partially answering a question posed by Hillel Furstenberg in 1972, we prove that if the action of $H$ on $G / Γ$ is minimal, then it is uniquely ergodic. Our proof uses in an essential way Marina Ratner’s classification of probability measures on $G / Γ$ invariant under unipotent elements, and the study of “tubes” in $G / Γ$ .

Generalized Staircases: Recurrence and Symmetry

W. Patrick Hooper; Barak Weiss — 2012

Annales de l’institut Fourier

We study infinite translation surfaces which are $ℤ$ -covers of compact translation surfaces. We obtain conditions ensuring that such surfaces have Veech groups which are Fuchsian of the first kind and give a necessary and sufficient condition for recurrence of their straight-line flows. Extending results of Hubert and Schmithüsen, we provide examples of infinite non-arithmetic lattice surfaces, as well as surfaces with infinitely generated Veech groups.

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