Exponential Decay of Correlations for Piecewise Cone Hyperbolic Contact Flows

Abstract

Abstract. We prove exponential decay of correlations for a realistic model of piecewise hyperbolic flows preserving a contact form, in dimension three. This is the first time exponential decay of correlations is proved for continuous-time dynamics with singularities on a manifold. Our proof combines the second author's version [30] of Dolgopyat's estimates for contact flows and the first author's work with Gouëzel [6] on piecewise hyperbolic discrete-time dynamics.

Anosov flows and dynamical zeta functions

“…The choice of stating the Lemma in terms of the norm · * η is a bit arbitrary but very convenient in the present context. The estimate is largely norm-independent as better shown in [8].…”

“…In the following we will restrict to the latter since it covers the geometrically relevant case of geodesic flows in negative curvature. Our approach follows roughly [42, Section 6] but employs several simplifying ideas, some from [8], some new.…”

“…To do so we use the simple strategy put forward in [44]. Finally, to establish the Dolgopyat estimate in the present context we follow the strategy developed in [42,43,8] for the action on functions. Note that to look at d s -forms corresponds, in the old Markov based strategy, to studying the statistical properties of the flow with respect to the measure of maximal entropy.…”

“…Unfortunately, in Section 7 of [16], the authors did not take into consideration that for t ≤ t 0 , φ t (W α,G ) is not necessarily controlled by the cones defined by (3.2), and thus could be an inadmissible manifold. Such an issue can be easily fixed, in the present context, by introducing a dynamical norm (similarly to [8]) as done below. This however is not suitable for studying perturbation theory, to this end a more radical change of the Banach space is required [17].…”

“…of short times we restrict ourself to a new space B p,q,ℓ which is the closure of Ω ℓ 0,s (M) with respect to a slightly stronger norm |||·||| p,q,ℓ (see (4.6)), in this we follow [8]. We easily obtain Lasota-Yorke type estimate for all times in the new norm (Lemma 4.7).…”