@MISC{Liverani93decayof, author = {Carlangelo Liverani}, title = {Decay of Correlations}, year = {1993} }

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Abstract

this paper I describe a technique, originally due to G. Birkhoff [9], [10], that permits a direct study of the Perron-Frobenius operator, and I show that its field of applicability is wider than that of Markov partitions. In essence, it is possible to construct systematically metrics (Hilbert metrics) with respect to which the Perron-Frobenius operator is a contraction. Such contraction allows to obtain the invariant measure (if not already known) by an elementary, and constructive, fixed point theorem, rather than by some compactness argument (this may please some idiosyncratic people, myself included), and automatically implies an exponential rate for the decay of correlations. I illustrate such an approach by applying it to several examples. For the sake of brevity and clarity the results are not presented in their full generality. DECAY OF CORRELATIONS 3 In particular, all the arguments used for two-dimensional smooth maps can be extended to the n-dimensional case. Results concerning more general systems (notably billiards and non-uniformly hyperbolic maps) will be published in separate papers. I also hope that the present exposition will prompt others to try to apply this method to the many cases where it could yield new results (e.g. dissipative systems, flows, etc.). The structure of the paper is as follows: section 1 describes the Hilbert metric and its properties. It is a brief review of the subject, intended to provide an easy reference for the reader. Section 2 shows how the technique works in the simplest example: a one dimensional uniformly hyperbolic map. It also mentions other consequences that can be obtained (e.g. Central Limit Theorem type results). In sections 3 I show how to extend the approach to the multidimensional case---the smooth case is trea...