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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Bibliographic Details
Published in:Journal of statistical physics Vol. 166; no. 3-4; pp. 525 - 557
Main Author: Baladi, Viviane
Format: Journal Article
Language:English
Published: New York Springer US 01.02.2017
Springer
Springer Nature B.V
Springer Verlag
Series:Special Issue: Dedicated to David Ruelle & Yakov Sinai
Subjects:
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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Summary: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.
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ISSN:0022-4715
1572-9613
DOI:10.1007/s10955-016-1663-0