Topologically ergodic C0-semigroups and ergodicity of their operators

This paper introduces topologically ergodic C0-semigroups. It demonstrates that the set of topologically ergodic C0-semigroups is a proper subset of the set of recurrent C0-semigroups. It also proves that these C0-semigroups exist in any infinite-dimensional space. In addition, the article establishes that the topological ergodicity of any operator of a0 C-semigroup implies the topological ergodicity of the0 C- semigroup and vice versa. Moreover, the research shows that the topological ergodicity of a0 C-semigroup is equivalent to the topological ergodicity of the direct sum of the0 C-semigroup with itself. Lastly, the research states some criteria for topological ergodicity that are in accordance with open sets and generalized kernels.