High frequency limits for invariant Ruelle densities
Annales Henri Lebesgue, Volume 4 (2021) , pp. 81-119.

KeywordsRuelle resonances, quantum ergodicity, semi-classical measures

### Abstract

We establish an equidistribution result for Ruelle resonant states on compact locally symmetric spaces of rank $1$. More precisely, we prove that among the first band Ruelle resonances there is a density one subsequence such that the respective products of resonant and co-resonant states converge weakly to the Liouville measure. We prove this result by establishing an explicit quantum-classical correspondence between eigenspaces of the scalar Laplacian and the resonant states of the first band of Ruelle resonances, which also leads to a new description of Patterson–Sullivan distributions.

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