Ergodicity of the finite and infinite dimensional $\alpha$-stable systems

Some finite and infinite dimensional perturbed -stable dynamics are constructed and studied in this article. We prove that the finite dimensional system is strongly mixing, while in the infinite dimensional case that the functional coercive inequalities are not available, we develop and apply a technique to prove the point-wise ergodicity for systems with sufficiently small interaction in a large subspace of O = RZd.