We present a new scale Upt,s​ (with s<−t<0 and 1≤p<∞) 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 Upt,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 ds​=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 Mb​ is compact.
Version v3 contains an additional text "Corrections and complements" showing
that s> t-(r-1) is needed in Section 4.Comment: 31 pages, revised version following referees' reports, with
Corrections and complement