Exponential decay of correlations for surface semiflows with an expanding direction

Dolgopyat [4] showed that a class of Axiom A flows has exponential decay of correlations for smooth observables, and Baladi-Vallée [2] gave a nice interpretation of it on suspension semiflows of one-dimensional expanding countable Markov maps. Avila-Gouëzel-Yoccoz [1] extends the result of Baladi-Vallée to higher dimensional systems. In this paper we show that a class of non-Markov semiflows also has exponential decay of correlations. We prove that such exponential decay can be shown on an open dense condition for the suspensions of piecewise expanding maps.