Decay of Correlations in Hyperbolic Systems

I illustrate a unified approach to the study of the decay of correlations in hyperbolic dynamical systems. The decay of correlations or, alternatively, the rate of approach of some initial distribution to an invariant one, is a widely studied problem in dynamical systems as well as in other fields. Although a full understanding is not yet available, some notable results have been obtained in the case of hyperbolic dynamical systems. The main strategy in deriving such results is the study of the transfer operator (also called Perron-Frobenius, or Ruelle-Perron-Frobenius, operator), either directly (as in the case of some one-dimensional systems) or indirectly (after coding the dynamics by introducing special partitions of the phase space called Markov partitions). I suggest a new approach to the study of the Perron-Frobenius operator, that avoids Markov partitions altogether, and permits a direct study of its properties also in the multidimensional case. Such a technique allows not only..