Fredholm determinants, Anosov maps and Ruelle resonances
By C. Liverani. 2005
61 citations recorded by the source corpus.
I show that the dynamical determinant, associated to an Anosov diffeomorphism, is the Fredholm determinant of the corresponding Ruelle-Perron-Frobenius transfer operator acting on appropriate Banach spaces. As a consequence it follows, for example, that the zeroes of the dynamical determinant describe the eigenvalues of the transfer operator and the Ruelle resonances and that, for $\C^\infty$ Anosov diffeomorphisms, the dynamical determinant is an entire function.
Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/04200a3b4c41ad953b960b399c2d1b87352abbef
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Geometry and Topology
In Mathematical Physics
- Viviane Baladi
- M. Tsujii
- Oliver Butterley
- C. Liverani
- David Ruelle
- S. Gouëzel
- Gary Froyland
- Mark Pollicott
In Statistical and Nonlinear Physics
- Viviane Baladi
- C. Liverani
- Oliver Butterley
- M. Tsujii
- David Ruelle
- S. Gouëzel
- B. Saussol
- Gary Froyland
Citations in the corpus
- Dynamical determinants and spectrum for hyperbolic diffeomorphisms (2006, methodology): “traces The second difficulty to solve this problem in the case of hyperbolic Cr diffeomorphisms is to find an appropriate definition for the trace of transfer operators that are not even compact. Liverani [19] found a simple argument to relate eigenvalues of L with zeroes of the dynamical Fredholm determ”
- Smooth Anosov flows: Correlation spectra and stability (2007, methodology): “Finally, it should be remarked that the approach of the present paper is based on the study of the resolvent, rather than the semigroup, in the spirit of [ 20 ].”