Ruelle-Pollicott resonances for real analytic hyperbolic map

Abstract

We study two simple real analytic uniformly hyperbolic dynamical systems: expanding maps on the circle S1 and hyperbolic maps on the torus T2. We show that the Ruelle-Pollicott resonances which describe time correlation functions of the chaotic dynamics can be obtained as the eigenvalues of a trace class operator in Hilbert space L2(S1) or L2(T2) respectively. The trace class operator is obtained by conjugation of the Ruelle transfer operator in a similar way quantum resonances are obtained in open quantum systems. We comment this analogy.

26 pages, 7 figures

References in corpus (3)

• Ruelle-Perron-Frobenius spectrum for Anosov maps
• The Selberg zeta function for convex co-compact Schottky groups
• Spectral properties of noisy classical and quantum propagators

Cited by in corpus (21)

• Upper bound on the density of Ruelle resonances for Anosov flows
• The quest for the ultimate anisotropic Banach space
• Complete spectral data for analytic Anosov maps of the torus
• Zeta functions and Dynamical Systems
• Generic non-trivial resonances for Anosov diffeomorphisms
• Eigenfunctions of the Perron-Frobenius operator and the finite-time Lyapunov exponents in uniformly hyperbolic area-preserving maps
• On the spectra of quenched random perturbations of partially expanding maps on the torus
• Fractal Weyl law for the Ruelle spectrum of Anosov flows
• FBI Transform in Gevrey Classes and Anosov Flows
• Explicit a priori bounds on transfer operator eigenvalues
• Resonance chains in open systems, generalized zeta functions and clustering of the length spectrum
• A local trace formula for Anosov flows (with an appendix by Frédéric Naud)
• Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps
• On the support of Pollicott-Ruelle resonant states for Anosov flows
• Micro-local analysis of contact Anosov flows and band structure of the Ruelle spectrum
• Explicit examples of resonances for Anosov maps of the torus
• Mathematical Study of Scattering Resonances
• Chaotic fields out of equilibrium are observable independent
• Heat kernel and the rate of mixing for compact extensions of expanding maps
• The spectral gap for transfer operators of torus extensions over expanding maps
• Explicit eigenvalue estimates for transfer operators