In a classical approach to dynamical systems, one frequently uses certain geometric structures (Markov partitions) which allow, under a codification of the system, to deduce certain statistical properties of the system, such as the existence of invariant measures with stochastic-like behaviour, large deviations or decay of correlations. In general, proving the existence of such geometric structures is a non-trivial problem. Thus, a natural question is the extent to which this approach can be applied. We show that, in many cases, stochastic-like behaviour itself implies the existence of certain geometric structures, which are therefore necessary as well as sufficient conditions for the occurrence of the statistical properties under consideration.

# Geometric structures vs. statistical properties in dynamical systems

José Ferreira Alves (CMUP)