Dolgopyat  showed that a class of Axiom A flows has exponential decay of correlations for smooth observables, and Baladi-Vallée  gave a nice interpretation of it on suspension semiflows of one-dimensional expanding countable Markov maps. Avila-Gouëzel-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.
Ippei Obayashi. "Exponential decay of correlations for surface semiflows with an expanding direction." J. Math. Kyoto Univ. 49 (2) 427 - 440, 2009. https://doi.org/10.1215/kjm/1256219166