  In this paper, we study the ergodicity of invariant sublinear expectation of
sublinear Markovian semigroup. For this, 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 regular. For
sublinear Markov process, we prove that its ergodicity is equivalent to the
Markovian semigroup having eigenvalue $1$ and it is simple in the space of
bounded measurable functions. As an example we show that $G$-Brownian motion
$\{B_t\}_{t\geq 0}$ on the unit circle has an invariant expectation and is
ergodic if and only if ${\mathbb E}(-(B_1)^2)<0$. Moreover, it is also proved
in this case that the invariant expectation is regular and the canonical
stationary process has no mean-uncertainty under the invariant expectation.
