A classic approach in dynamical systems is to use particular geometric
structures to deduce statistical properties, for example the existence of
invariant measures with stochastic-like behaviour such as large deviations or
decay of correlations. Such geometric structures are generally highly
non-trivial and thus a natural question is the extent to which this approach
can be applied. In this paper we show that in many cases stochastic-like
behaviour itself implies that the system has certain non-trivial geometric
properties, which are therefore necessary as well as sufficient conditions for
the occurrence of the statistical properties under consideration. As a by
product of our techniques we also obtain some new results on large deviations
for certain classes of systems which include Viana maps and multidimensional
piecewise expanding maps.