Abstract. 
We consider hyperbolic systems with general singularities 
including chaotic billiards. In order to uncover the fact that 
hyperbolicity overcomes the fragmentation by singularities, we obtain 
the Growth lemma  under weaker assumptions than previously adopted
in similar studies. Furthermore,  proper measures were coupled whenever 
both images under the map simultaneously attracted to a "magnet".   
Our approach allows us to obtain the exponential decay of correlations 
for a larger class of billiards.