Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms

Abstract

We study spectral properties of transfer operators for diffeomorphisms T : X → X on a Riemannian manifold X: Suppose that Ω is an isolated hyperbolic subset for T , with a compact isolating neighborhood V ⊂ X. We first introduce Banach spaces of distributions supported on V , which are anisotropic versions of the usual space of C p functions C p (V ) and of the generalized Sobolev spaces W p,t (V ), respectively. Then we show that the transfer operators associated to T and a smooth weight g extend boundedly to these spaces, and we give bounds on the essential spectral radii of such extensions in terms of hyperbolicity exponents. * (T, V ), for 1 < t < ∞, of distributions supported on V , such that, for any C r−1 function g : X → C supported on V ,(1) (Hölder spaces) L T,g extends boundedly to L T,g : C p,q * (T, V ) and r ess (L T,g | C p,q * (T,V ) ) ≤ R p,q,∞ (T, g, Ω). (2) (Sobolev spaces) L T,g extends boundedly to L T,g : W p,q,t *

Anosov flows and dynamical zeta functions

“…Since then a considerable amount of work has been done to implement such a goal. In particular, [30,44,9,31] and especially [10] have clarified the relation between the smoothness of the map and the essential spectral radii of transfer…”

“…. , ω α,d be the dual basis 9 The relations are meant to be valid where the composition is defined. The · C r norm is precisely defined in (3.6).…”

Exponential Decay of Correlations for Piecewise Cone Hyperbolic Contact Flows

“…This approach was developed in the next few years for smooth discrete-time hyperbolic dynamics by Baladi [3], Gouëzel-Liverani [23]- [24], and Baladi-Tsujii [7]- [8], and more recently for some smooth hyperbolic flows (Butterley-Liverani [12], Tsujii [45] [46]). Except for the Sobolev-Triebel spaces used in [3] (where a strong assumption of regularity of the dynamical foliation was required), it turns out that the Banach spaces appropriate for smooth hyperbolic dynamics are not suitable for systems with discontinuities, because multiplication by the characteristic function of a domain, however nice, is not a bounded operator for the corresponding norms.…”

Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow

“…If B is carefully tuned (its elements should be smooth in the stable direction of the flow, and dual of smooth in the unstable direction), then one can hope to get smoothing effects in every direction, and therefore some compactness. This kind of arguments has been developed in recent years for Anosov maps or flows in compact manifolds and has proved very fruitful (see among others [Liv04,GL06,BT07,BL07]). We use in an essential way the insights of these papers.…”