The Quest for the Ultimate Anisotropic Banach Space

We present a new scale $U_p^{t,s}$ ($s<-t<0$ and $1\le p<\infty$) of anisotropic Banach spaces, defined via Paley–Littlewood, on which the transfer operator $L_g\phi =(g\cdot \phi )\circ T^{-1}$ 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.