Exponential decay of correlations for piecewise cone hyperbolic contact flows

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 of Dolgopyat's estimates for contact flows and the first author's work with Gou\"ezel on piecewise hyperbolic discrete-time dynamics. (Presentation revised.)