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.
"Decay of Correlations for Non-Hölderian Dynamics. A Coupling Approach." Electron. J. Probab. 4 1 - 19, 1999. https://doi.org/10.1214/EJP.v4-40