In my talk, I will rst talk an ergodic theory of an expectation-preserving map on a sublinear expectation space. We also study the ergodicity of invariant sublinear expectation of sublinear Markovian semigroup. As an example we show that G-Brownian motion on the unit circle has an invariant expectation and is ergodic. In the last part of my talk, I will discuss the ergodic theory of Invariant capacity.

This is a joint work with Huaizhong Zhao.