The Lipschitz ergodicity and generalized ergodicity of semigroups

In this paper, the semigroup is defined by the continuous transformation on the compact metric space. We use the chain recurrence to study the Lipschitz ergodicity and generalized ergodicity. Some criterions for a system to be Lipschitz ergodic or generalized ergodic are given. At last, we illustrate the relationship between ergodicity, generalized ergodicity and Lipschitz ergodicity.