  In this article, we study the decay rates of the correlation functions for a
hyperbolic system $T: M \to M$ with singularities that preserves a unique
mixing SRB measure $\mu$. We prove that, under some general assumptions, the
correlations $ C_{n}(f,g)$ decay exponentially as $n\to \infty$ for each pair
of piecewise H\"older observables $f, g\in L^p(\mu)$ and for each $p>1$. As an
application, we prove that the autocorrelations of the first return time
functions decay exponentially for the induced maps of various billiard systems,
which include the semi-dispersing billiards on a rectangle, billiards with
cusps, and Bunimovich stadia (for the truncated first return time functions).
These estimates of the decay rates of autocorrelations of the first return time
functions for the induced maps have an essential importance in the study of the
statistical properties of nonuniformly hyperbolic systems (with singularities).
