The quest for the ultimate anisotropic Banach space

TL;DR

This paper introduces a new family of anisotropic Banach spaces tailored for hyperbolic dynamical systems, enabling better spectral analysis and connections to dynamical determinants and zeta functions.

Contribution

It defines a novel scale of anisotropic Banach spaces $U^{t,s}_p$ with specific spectral properties, extending previous geometric and microlocal frameworks for dynamical systems analysis.

Findings

Spaces have good spectral properties for transfer operators.

Spaces relate to geometric and microlocal approaches.

The paper proves bounds on the essential spectral radius.

Abstract

We present a new scale $U^{t,s}_p$ (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 Gou\"ezel 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 $U^{t,s}_p$ 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 $\cal M_b$ is compact. Version v3 contains an additional text "Corrections and complements" showing that s> t-(r-1) is needed in Section 4.