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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 Hölder 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. Abstract We present an upper bound on the mixing rate of the equilibrium state of a dynamical system defined by thedoi:10.1214/ejp.v4-40 fatcat:njmspjpb45fr5dddgkrjwqudyu