×

zbMATH — the first resource for mathematics

Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. (French. English summary) Zbl 1138.37011
Summary: We study spectral properties of transfer operators for diffeomorphisms \(T:X \to X\) on a Riemannian manifold \(X\). Suppose that \(\Omega \) is an isolated hyperbolic subset for \(T\), with a compact isolating neighborhood \(V \subset X\). We first introduce Banach spaces of distributions supported on \(V\), which are anisotropic versions of the usual space of \(C^{ p }\) functions \(C^{ p }(V)\) and of the generalized Sobolev spaces \(W^{ p,t }(V)\), respectively. We then show that the transfer operators associated to \(T\) and a smooth weight \(g\) extend boundedly to these spaces, and we give bounds on the essential spectral radii of such extensions in terms of hyperbolicity exponents.

MSC:
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML arXiv
References:
[1] Avila, A.; Gouëzel, S.; Tsujii, M., Smoothness of solenoidal attractors, Discrete Cont. Dynam. Systems, 15, 21-35, (2006) · Zbl 1106.37015
[2] Baladi, V., Advanced Series in Nonlinear Dynamics, 16, Positive transfer operators and decay of correlations, (2000), World Scientific · Zbl 1012.37015
[3] Baladi, V., Algebraic and Topological Dynamics, Contemporary Mathematics, Anisotropic Sobolev spaces and dynamical transfer operators:\( C^{∞ }\) foliations, S.kolyada, Y.Manin and T.Ward, eds., 123-136, (2005), Amer. Math. Soc. · Zbl 1158.37304
[4] Blank, M.; Keller, G.; Liverani, C., Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15, 1905-1973, (2002) · Zbl 1021.37015
[5] Fried, D., The flat-trace asymptotics of a uniform system of contractions, Ergodic Theory Dynam. Sys., 15, 1061-1073, (1995) · Zbl 0841.58052
[6] Fried, D., Meromorphic zeta functions for analytic flows, Comm. Math. Phys., 174, 161-190, (1995) · Zbl 0841.58053
[7] Gouëzel, S.; Liverani, C., Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Sys., 26, 189-218, (2006) · Zbl 1088.37010
[8] Gundlach, V. M.; Latushkin, Y., A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Hölder spaces, Ergodic Theory Dynam. Sys., 23, 175-191, (2003) · Zbl 1140.37307
[9] Hennion, H., Sur un théorème spectral et son application aux noyaux lipschitziens, Proc. Amer. Math. Soc., 118, 627-634, (1993) · Zbl 0772.60049
[10] Hörmander, L., Grundlehren der Mathematischen Wissenschaften, 274, The analysis of linear partial differential operators. III. pseudo-differential operators, (1994), Springer-Verlag, Berlin · Zbl 0601.35001
[11] Kitaev, A. Yu., Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness, Nonlinearity, 12, 141-179, (1999) · Zbl 0917.58029
[12] Paley, J. E.; Littlewood, R., Theorems on Fourier series and power series, Proc. London Math. Soc., 42, 52-89, (1937) · Zbl 0015.25402
[13] Ruelle, D., The thermodynamic formalism for expanding maps, Comm. Math. Phys., 125, 239-262, (1989) · Zbl 0702.58056
[14] Rugh, H. H., The correlation spectrum for hyperbolic analytic maps, Nonlinearity, 5, 1237-1263, (1992) · Zbl 0768.58027
[15] Taylor, M. E., Lecture Notes in Math., 416, Pseudo differential operators, (1974), Springer-Verlag, Berlin-New York · Zbl 0289.35001
[16] Taylor, M. E., Progress in Math., 100, Pseudodifferential operators and nonlinear PDE, (1991), Birkhäuser, Boston · Zbl 0746.35062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.