  We consider invariant measures of maps on manifolds whose correlations decay
at a sufficient rate and which satisfy a geometric contraction property. We
then prove the that the limiting distribution of returns to geometric balls is
Poissonian. This does not assume an tower construction. The decay of
correlations is used to show that the independence generated results in the
Poisson distribution for returns that are sufficiently separated. A geometric
contraction property is then used to show that short return times have a
vanishing contribution to the return times distribution. We then also show that
the set of very short returns which are of a small linear order of the
logarithm of the radius of the balls has a vanishing measure. We obtain error
terms which decay polynomially in the logarithm of the radius. We also obtain a
extreme value law for such systems.
