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.