The quest for the ultimate anisotropic Banach space

Abstract

We present a new scale $U_p^{t,s}$ (with $s<-t<0$ and $1\le p<\infty$) of anisotropic Banach spaces, defined via Paley-Littlewood, on which the transfer operator associated to a hyperbolic dynamical system has good spectral properties. When $p=1$ and $t$ is an integer, the spaces are analogous to the "geometric" spaces considered by Gouëzel and Liverani. 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. In addition, just like for the "microlocal" spaces defined by Baladi-Tsujii, the spaces $U_p^{t,s}$ are amenable to the kneading approach of Milnor-Thurson to study dynamical determinants and zeta functions. In v2, following referees' reports, typos have been corrected (in particular (39) and (43)). Section 4 now includes a formal statement (Theorem 4.1) about the essential spectral radius if $d_s=1$ (its proof includes the content of Section 4.2 from v1). The Lasota-Yorke Lemma 4.2 (Lemma 4.1 in v1) includes the claim that ${\mathcal{M}}_b$ is compact. Version v3 contains an additional text "Corrections and complements" showing that s> t-(r-1) is needed in Section 4.

31 pages, revised version following referees' reports, with Corrections and complements

References in corpus (4)

• The quest for the ultimate anisotropic Banach space
• Anisotropic Sobolev spaces and dynamical transfer operators: C^infty foliations
• Zeta functions and Dynamical Systems
• Anisotropic, Mixed-Norm Lizorkin--Triebel Spaces and Diffeomorphic Maps

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