Ruelle-Perron-Frobenius spectrum for Anosov maps
Blank M, Keller G, Liverani C (2002)
Publication Type: Journal article, Original article
Publication year: 2002
Journal
Publisher: Institute of Physics: Hybrid Open Access
Book Volume: 15
Pages Range: 1905-1973
Journal Issue: 6
DOI: 10.1088/0951-7715/15/6/309
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 us 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 quasi-compact. (Information on the existence of a Sinai-Ruelle-Bowen measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension d = 2 we show that the transfer operator associated with smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows us 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.
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APA:
Blank, M., Keller, G., & Liverani, C. (2002). Ruelle-Perron-Frobenius spectrum for Anosov maps. Nonlinearity, 15(6), 1905-1973. https://doi.org/10.1088/0951-7715/15/6/309
MLA:
Blank, Michael, Gerhard Keller, and Carlangelo Liverani. "Ruelle-Perron-Frobenius spectrum for Anosov maps." Nonlinearity 15.6 (2002): 1905-1973.
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