High frequency limits for invariant Ruelle densities

Guillarmou, Colin; Hilgert, Joachim; Weich, Tobias

Annales Henri Lebesgue, Volume 4 (2021), pp. 81-119.

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.

On établit un résultat d’équidistribution pour les états résonants de Ruelle sur les espaces localement symétriques compacts de rang $1$. Plus précisemment, on montre que, parmi les résonances de Ruelle dans la première bande, il y a une sous-suite de densité $1$ pour laquelle le produit des états résonants et co-résonants associés converge faiblement vers la mesure de Liouville. On démontre ce résultat via l’obtention d’une correspondance exacte entre les espaces propres du Laplacien et les états résonants de Ruelle de la première bande, ce qui par ailleurs donne une nouvelle description des distributions de Patterson–Sullivan.

Classification: 37D20, 53C35, 35P20
Mots-clés : Ruelle resonances, quantum ergodicity, semi-classical measures

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