Dolgopyat  showed that a class of Axiom A flows has exponential decay of correlations for smooth observables, and Baladi-Vallee  gave a nice interpretation of it on suspension semiflows of one-dimensional expanding countable Markov maps. Avila-Goue7die;zel-Yoccoz  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.
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