Abstract
We classify measures on the locally homogeneous space $\Gamma \backslash SL(2,\mathbb{R}) \times L$ which are invariant and have positive entropy under the diagonal subgroup of $SL(2,\mathbb{R})$ and recurrent under $L$. This classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces, and a similar but slightly weaker result for the finite volume case. Other applications are also presented.
In the appendix, joint with D. Rudolph, we present a maximal ergodic theorem, related to a theorem of Hurewicz, which is used in the proof of the main result.
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