Ruelle Perron Frobenius spectrum for Anosov maps

Abstract

Abstract. We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension d = 2 we show that the transfer operator associated to smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows to obtain easily very strong spectral stability results, which in turn imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite dimensional problem.

Spectral analysis of mixing in chaotic flows via the mapping matrix formalism: Inclusion of molecular diffusion and quantitative eigenvalue estimate in the purely convective limit

“…In the mathematically oriented literature, Ulam's method and its convergence properties has been first tackled by Li 37 that proved the convergence of the Ulam's method to the unique absolutely continuous invariant measure for a class of interval maps, and subsequently by many other authors 19,20,[38][39][40][41] that extended the analysis to higher dimensional dynamical systems and applied Ulam's method to determine entropies and other dynamical invariants. 42 It is rather remarkable that these two lines of research essentially focused on the same topic, that can be referred to in a "fair" way as the Spencer-Wiley-Ulam method to study dynamical systems using a coarse-grained representation of the evolution equation for densities has proceeded for 073603-7…”

Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms

“…Spectral stability can then be proved, as it has been done [4] or [7] for the norms defined there (see also the historical comments below).…”

On contact Anosov flows

“…An improvement on the error term for the above formula, that can be found in this paper (Appendix B, Lemma B.7), shows that, for a contact Anosov flow, if the strong foliations are τ -Hölder, with τ > √ 3−1, then the temporal function is likely to be C 1 (see Remark B.8). On the other hand, geodesic flows that are a-pinched 2 have foliations that are C 2 √ a ( [18] and Appendix B; see also [13], [11] for more complete results on such an issue). Dolgopyat's results would then, at best, imply that any geodesic flow in negative curvature which is a-pinched, with a > 1 − √ 3 2 , enjoys exponential decay of correlations.…”

“…To obtain such a result I built on Dolgopyat's work and on the results in [2] where a functional space is introduced over which the Perron-Frobenius operator can be studied directly, without any coding, contrary to the previous approaches by Dolgopyat, Chernov and Pollicott.…”