The Quest for the Ultimate Anisotropic Banach Space
نویسنده
چکیده
We present a new scale U p (s < −t < 0 and 1 ≤ p < ∞) of anisotropic Banach spaces, defined via Paley–Littlewood, on which the transfer operator Lgφ = (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 Bt,|s+t| considered by Gouëzel and Liverani [26]. 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 [16]. In addition, just like for the “microlocal” spaces defined by Baladi–Tsujii [10] (or Faure–Roy–Sjöstrand [19]), the transfer operator acting on U p can be decomposed into Lg,b+Lg,c, where Lg,b has a controlled norm while a suitable power of Lg,c is nuclear. This “nuclear power decomposition” enhances the Lasota–Yorke bounds and makes the spaces U p amenable to the kneading approach of Milnor–Thurson [34] (as revisited by Baladi–Ruelle [8, 9, 2]) to study dynamical determinants and zeta functions.
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