Smooth Anosov flows: Correlation spectra and stability

Exponential Decay of Correlations for Piecewise Cone Hyperbolic Contact Flows

“…Lemma 3.5 implies that the spectrum of R(z) outside of the disk {|ρ| ≤ (a + ln(1/λ)) −1 }, for ℜz = a > A, consists only of isolated eigenvalues of finite multiplicity. Let us assume that there is a unique maximal eigenvalue ρ(z) (the case of finitely many maximal eigenvalues is treated in exactly the same way, apart for 12 Remember that R(z) is a well-defined operator both on L ∞ and L 1 , abusing notation we use the same name for the operator defined on different spaces. the necessity of a heavier notation).…”

“…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.…”

Anosov flows and dynamical zeta functions

“…On the other hand, [43,16] illustrated how this approach can be applied also to flows by showing that the resolvent of the generator of the flow is quasi compact on such spaces. More recently some beautiful work, although limited to the case of contact flows, allows one to study directly the transfer operator associated to the time one map of the flow [73,74,25,24].…”

Linear response and periodic points

“…Contributions to the study of this and related problems have been made by Ruelle [15] and [16], Baladi and Smania [3], [4], [5]; Dolgopyat [9], Liverani and Butterley [6].…”