The quest for the ultimate anisotropic Banach space

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.