Decay of correlations. (English) Zbl 0871.58059

Decays of correlations, and in particular the question of an exponential rate for the decay, are studied both in statistical mechanics by studying the Ruelle-Perron Frobenius operator or in symbolic dynamics, by coding hyperbolic dynamical systems via Markov partitions. This paper shows that the correlations decay exponentially fast for a general class of piecewise smooth hyperbolic maps.
The technique used here is inspired by G. Birkhoff [Trans. Am. Math. Soc. 85, 219-227 (1957; Zbl 0079.13502); ‘Lattice theory’, 3rd ed. (1967; Zbl 0153.02501)].
The transfer operator is shown to be a contraction with respect to an explicitly constructed Hilbert metric, which provides constructively the invariant measure.
The results can be extended to a more general class of hyperbolic maps such as billiards and non-uniformly hyperbolic maps, and could possibly provide new ways of approach for dissipative systems, for instance.


37A99 Ergodic theory
37D99 Dynamical systems with hyperbolic behavior
47B38 Linear operators on function spaces (general)
Full Text: DOI Link