In this paper, we first develop an ergodic theory of an expectation-preserving map
on a sublinear expectation space. Ergodicity is defined as any invariant set either
has $0$ capacity itself or its complement has $0$ capacity. We prove, under a general
sublinear expectation space setting, the equivalent relation between ergodicity and the
corresponding transformation operator having simple eigenvalue $1$, and also
with Birkhoff type strong law of large numbers if the sublinear expectation is strongly regular.
We also study the ergodicity of invariant sublinear expectation of sublinear Markovian semigroup.
We prove that its ergodicity is equivalent to the generator of the Markovian semigroup having
eigenvalue $0$ and the eigenvalue is simple in the space of continuous functions. As an example
we show that $G$-Brownian motion on the unit circle has an invariant expectation and is ergodic.
Moreover, it is also proved in this case that the invariant expectation is strongly regular and the
canonical stationary process has no mean-uncertainty under the invariant expectation.
This is a joint work with Chunrong Feng.