The Quest for the Ultimate Anisotropic Banach Space

We present a new scale U p t , s ( s < - t < 0 and 1 ≤ p < ∞ ) of anisotropic Banach spaces, defined via Paley–Littlewood, on which the transfer operator L g φ = ( g · φ ) ∘ T - 1 associated to a hyperbolic dynamical system T has good spectral properties. When p = 1 and t is an integer, the...

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Vydáno v:	Journal of statistical physics Ročník 166; číslo 3-4; s. 525 - 557
Hlavní autor:	Baladi, Viviane
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.02.2017 Springer Springer Nature B.V Springer Verlag
Edice:	Special Issue: Dedicated to David Ruelle & Yakov Sinai
Témata:	ANISOTROPY BANACH SPACE Banach spaces CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS DECOMPOSITION DYNAMICAL SYSTEMS GEOMETRY Mathematical and Computational Physics Mathematics Nuclear energy NUCLEAR POWER Operators (mathematics) Physical Chemistry Physics Physics and Astronomy Quantum Physics Statistical Physics and Dynamical Systems Theoretical Piecewise Smooth Systems Essential Spectral Radius 2-microlocal Spaces Bunching Conditions Anisotropic Space hyperbolic dynamics transfer operator spectrum anisotropic Banach space
ISSN:	0022-4715, 1572-9613
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Popis
Shrnutí:	We present a new scale U p t , s ( s < - t < 0 and 1 ≤ p < ∞ ) of anisotropic Banach spaces, defined via Paley–Littlewood, on which the transfer operator L g φ = ( g · φ ) ∘ T - 1 associated to a hyperbolic dynamical system T has good spectral properties. When p = 1 and t is an integer, the spaces are analogous to the “geometric” spaces B t , \| s + t \| considered by Gouëzel and Liverani (Ergod Theory Dyn Syst 26:189–217, 2006 ). When p > 1 and - 1 + 1 / p < s < - t < 0 < t < 1 / p , the spaces are somewhat analogous to the geometric spaces considered by Demers and Liverani (Trans Am Math Soc 360:4777–4814, 2008 ). In addition, just like for the “microlocal” spaces defined by Baladi and Tsujii (Ann Inst Fourier 57:127–154, 2007 ) (or Faure–Roy–Sjöstrand in Open Math J 1:35–81, 2008 ), the transfer operator acting on U p t , s can be decomposed into L g , b + L g , c , where L g , b has a controlled norm while a suitable power of L g , c is nuclear. This “nuclear power decomposition” enhances the Lasota–Yorke bounds and makes the spaces U p t , s amenable to the kneading approach of Milnor–Thurson (Dynamical Systems (Maryland 1986–1987), Springer, Berlin, 1988 ) (as revisited by Baladi–Ruelle, Baladi in Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, Monograph, 2016 ; Baladi and Ruelle in Ergod Theory Dyn Syst 14:621–632, 1994 ; Baladi and Ruelle in Invent Math 123:553–574, 1996 ) to study dynamical determinants and zeta functions.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0022-4715 1572-9613
DOI:	10.1007/s10955-016-1663-0