First band of Ruelle resonances for contact Anosov flows in dimension 3

2020/11/11 by Guillarmou, Colin, Cekic, Mihajlo · 1 citation
#37D20 #81Q20 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2011.05959

Abstract

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 | \rm Re(s)∈ [-ν_min+ε,-(1)/(2)ν_max-ε]∪ [-(1)/(2)ν_min+ε,0]\ for all ε>0, where 0ν_max/2). We also show polynomial bounds in s for the resolvent (-X-s)^-1 as |\rm 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 \citeFaTs1,FaTs2,FaTs3, using that dim E_u=dim E_s=1.

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