Prequantum transfer operator for Anosov diffeomorphism. (Preliminary Version).

Frédéric Faure (1) , Masato Tsujii (2)

1 IF - Institut Fourier
2 Department of Mathematics

This is a preliminary version and some other results will appear in the next version. We define the prequantization of a symplectic Anosov diffeomorphism, which is a U(1) extension of the diffeomorphism preserving an associated specific connection. We study the spectrum of the associated transfer operator, called prequantum transfer operator, restricted to the N-th Fourier mode with respect to the U(1) action on P. We investigate the spectral property in the limit N to infinity, regarding the transfer operator as a Fourier integral operator and using semi-classical analysis. In the main result, we show a " band structure " of the spectrum, that is, the spectrum is contained in a few separated annuli and a disk concentric at the origin.

Mots clés: Anosov diffeomorphism; Transfer operator; prequantization; geometric quantization; Semiclassical analysis; Ruelle resonances

Domaines: Physique mathématique [math-ph]; Systèmes dynamiques [math.DS]; Théorie spectrale [math.SP]

Frédéric Faure, Masato Tsujii. Prequantum transfer operator for Anosov diffeomorphism. (Preliminary Version).. 2012. ⟨hal-00703234v1⟩