FIRST BAND OF RUELLE RESONANCES FOR CONTACT ANOSOV FLOWS IN DIMENSION 3

Mihajlo Cekić (1, 2) , Colin Guillarmou (1, 2)

1 Université Paris-Saclay
2 LMO - Laboratoire de Mathématiques d'Orsay

We show, using semiclassical measures and unstable derivatives, that a smooth vector field X generating a contact Anosov flow on a 3-dimensional manifold M has only finitely many Ruelle resonances in the vertical strips {s ∈ C | Re(s) ∈ [−\nu_min + ε, − 1/2 \nu_max − ε] ∪ [− 1/2 \nu_min + ε, 0]} for all ε > 0, where 0 < \nu_min ≤ \nu_max are the minimal and maximal expansion rates of the flow (the first strip only makes sense if \nu_min > \nu_max/2). We also show polynomial bounds in s for the resolvent (−X − s)^{−1} as |Im(s)| → ∞ in Sobolev spaces, and obtain similar results for cases with a potential. This is a short proof of a particular case of the results by Faure-Tsujii in [FaTs0, FaTs2, FaTs3], using that dim Eu = dim Es = 1.

Domaines: Mathématiques [math]

Mihajlo Cekić, Colin Guillarmou. FIRST BAND OF RUELLE RESONANCES FOR CONTACT ANOSOV FLOWS IN DIMENSION 3. Communications in Mathematical Physics, 2021, 386, pp.1289--1318. ⟨10.1007/s00220-021-04090-2⟩. ⟨hal-03020967v2⟩