Decay of correlations. (English) Zbl 0871.58059

Liverani, Carlangelo

Ann. Math. (2) 142, No. 2, 239-301 (1995).

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.

Reviewer: V.Berthé (Marseille)

MSC:

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

Keywords:

decay of correlation; statistical mechanics; Ruelle-Perron Frobenius operator; symbolic dynamics; hyperbolic dynamical systems; Markov partitions; billiards; non-uniformly hyperbolic maps; dissipative systems

Citations:

Zbl 0079.13502; Zbl 0153.02501