Quenched Decay of Correlations for One-dimensional Random Lorenz Maps

We study rates of mixing for small random perturbations of one-dimensional Lorenz maps. Using a random tower construction, we prove that, for Hölder observables, the random system admits exponential rates of quenched correlation decay.

Decay of correlations for weakly expanding dynamical systems with Dini potentials under optimal quasi-gap condition

We find an optimal quasi-gap condition for a weakly expanding dynamical system associated with Dini potential. Under this optimal quasi-gap condition, we prove the Ruelle operator theorem and further the decay of the correlations for any weakly expanding dynamical systems with Dini potentials.

Mixing rates for potentials of non-summable variations

Mixing rates, relaxation rates, and decay of correlations for dynamics defined by potentials with summable variations are well understood, but little is known for non-summable variations. This paper exhibits upper bounds for these quantities for dynamics defined by potentials with square-summable variations. We obtain these bounds as corollaries of a new block coupling inequality between pairs of dynamics starting with different histories. As applications of our results, we prove a new weak invariance principle and a Hoeffding-type inequality.