The flat-trace asymptotics of a uniform system of contractions

Fried, David

doi:10.1017/s0143385700009792

Cited by 15 publications

(18 citation statements)

References 11 publications

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“…-To study spectral properties of Perron-Frobenius operators (and Ruelle transfer operators, more generally), it is primarily important to find appropriate spaces for them to act on. For C r expanding dynamical systems (r > 1) and C r−1 weights, Ruelle [13], and later Fried [5] and Gundlach-Latushkin [8], showed that the Banach space of C r−1 functions worked nicely. For Anosov diffeomorphisms usual function spaces do not work.…”

Section: Spectral Stabilitymentioning

confidence: 99%

Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms

Baladi¹,

Tsujii²

2007

Ann. inst. Fourier

145

249

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We study spectral properties of transfer operators for diffeomorphisms T : X → X on a Riemannian manifold X: Suppose that Ω is an isolated hyperbolic subset for T , with a compact isolating neighborhood V ⊂ 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. Then we 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. * (T, V ), for 1 < t < ∞, of distributions supported on V , such that, for any C r−1 function g : X → C supported on V ,(1) (Hölder spaces) L T,g extends boundedly to L T,g : C p,q * (T, V ) and r ess (L T,g | C p,q * (T,V ) ) ≤ R p,q,∞ (T, g, Ω). (2) (Sobolev spaces) L T,g extends boundedly to L T,g : W p,q,t *

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Section: Spectral Stabilitymentioning

confidence: 99%

Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms

Baladi¹,

Tsujii²

2007

Ann. inst. Fourier

145

249

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“…Gallavotti (1976) then found a differentiable Axiom A flow whose Ruelle dynamical zeta function ζ(s) had a non-polar singularity. Much more recently Fried (1995b) proved, combining Grothendieck techniques from the pioneering article of Ruelle (1976b) with novel ideas of Rugh (1994), that the dynamical zeta function of a real analytic Axiom A flow (without assuming smoothness of the stable and unstable bundles) could indeed be extended meromorphically to C (see Theorem 4.1 later).…”

Section: Introductionmentioning

confidence: 99%

Periodic orbits and dynamical spectra (Survey)

Baladi¹

1998

Ergod. Th. Dynam. Sys.

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“…Let A be a transfer operator in C(X ) and let A * : C * (X ) → C * (X ) be the adjoint operator to A. The formulae (28) and (29) can be rewritten in the following way:…”

Section: A B Antonevich Et Almentioning

confidence: 99%

Ont-entropy and variational principle for the spectral radii of transfer and weighted shift operators

Antonevich

Bakhtin

Лебедев

2010

Ergod. Th. Dynam. Sys.

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The paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations and the principal results have been achieved in the situation when the dynamical system is either reversible or a topological Markov chain. As the main summands, these principles contain the integrals over invariant measures and the Kolmogorov–Sinai entropy. In the paper we derive the variational principle for anarbitrarydynamical system. It gives the explicit description of the Legendre dual object to the spectral potential. It is shown that in general this principle contains not the Kolmogorov–Sinai entropy but a new invariant of entropy type—thet-entropy.

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The flat-trace asymptotics of a uniform system of contractions

Cited by 15 publications

References 11 publications

Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms

Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms

Periodic orbits and dynamical spectra (Survey)

Ont-entropy and variational principle for the spectral radii of transfer and weighted shift operators

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