Decay of Correlations in Hyperbolic Systems

I illustrate a uniied 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 elds. 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 multi-dimensional case. Such a technique allows not only to reproduce, in a more direct way, most of the known results but also to obtain new ones (e.g., exponential decay of correlations for a wide class of discontinuous maps). In essence, it is possible to construct metrics (Hilbert metrics) with respect to which the Perron-Frobenius operator is a contraction. Such a contraction allows one to obtain the invariant measure (if not already known) by an elementary, and constructive, xed point theorem, and automatically implies an exponential rate for the decay of correlations. The Hilbert metric is naturally deened on convex sets and is of interest in hyperbolic geometry (e.g., the disk equipped with the Hilbert metric is the Poincar e disk). If e C is a closed convex set in a topological space V then, given x; y 2 e C, one can deene the line going thruough x and y and call u; v the two intersections of such a line with @ e C (set them equal to 1 if there are no intersection). The Hilbert metric in e C is deened as (x; y) = ln jx ? ujjy ? vj jx ? vjjy ? uj : That is, (x; y) is the logarithm of the cross ratio of the points x; y; u; v.