The Quest for the Ultimate Anisotropic Banach Space - Baladi, Viviane (Journal of statistical physics, 2017)
We present a new scale U p t , s ( s < - t < 0 and 1 ≤ p < ∞ ) of anisotropic Banach spaces, defined via Paley–Littlewood, on which the transfer operator L g φ = ( g · φ ) ∘ 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.