Abstract
We present an upper bound on the mixing rate of the equilibrium state of a dynamical system defined by the one-sided shift and a non Holder potential of summable variations. The bound follows from an estimation of the relaxation speed of chains with complete connections with summable decay, which is obtained via a explicit coupling between pairs of chains with different histories.
Highlights
Let μφ be the equilibrium state associated to the continuous function φ
Previous approaches to the study of the mixing properties of the one-sided shift rely on the use of the transfer operator Lφ, defined by the duality, f ◦ T n g dμφ = f Lnφg dμφ
To estimate the mixing rate, Kondah, Maume and Schmitt (1996) proved first that the operator is contracting in the Birkhoff projective metric, while Pollicott (1997), following Liverani (1995), considered the transfer operator composed with conditional expectations
Summary
Let μφ be the equilibrium state associated to the continuous function φ. Our approach is based on a probabilistic interpretation of the duality (1.2) in terms of expectations, conditioned with respect to the past, of a chain with complete connections The convergence (1.1) is related to the relaxation properties of this chain In this paper, such relaxation is studied via a coupling method. Instead of letting the trajectories evolve independently, one can couple them from the beginning, reducing the “meeting time” and, obtaining a better rate of convergence (leading to the so-called Dobrushin’s ergodic coefficient) Doeblin published his results in a hardly known paper in the Revue Mathematique de l’Union Interbalkanique. In Bressaud, Fernandez, Galves (1997), the coupling approach was generalized to treat chains with complete connections These processes, introduced by Doeblin and Fortet (1937) (see Lalley, 1986) appear in a natural way in the context of dynamical systems.
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