Mixing of capacity preserving dynamical systems

The purpose of this study is to probe the transformations that preserve capacities, their ergodicity and mixing behaviors. Firstly, definitions of different levels of mixing and ergodicity are introduced. Then attention is paid to continuous transformations on topological spaces with invariant capacities on Borel s-algebra. Themain results include that strong mixing implies ergodicity and weak mixing implies weak ergodicity. Moreover, limit properties of mixing capacity preserving dynamical systems and connections between a capacity preserving dynamical system and its product system are also discussed. The novelty of this study is that we present several topological characterizations of ergodic and mixing capacity preserving dynamical systems.