In a couple of books I'm reading chapters devoted to the Ergodic theory. As a setting:
$(\Omega,\mathcal F,\mathsf P)$ is a probability space,
$X:(\Omega,\mathcal F)\to (S,\mathcal S)$ is random element and
$\varphi:(\Omega,\mathcal F)\to (\Omega,\mathcal F)$ is a measure-preserving map, i.e. for any $A\in \mathcal F$ it holds that $\mathsf P(\varphi^{-1} A) = \mathsf P(A).$
The event is called invariant if $\mathsf P((\varphi^{-1}A)\Delta A) = 0$. I am interested in some examples for such events since books I'm reading have a very few of them.
If we put $X_n(\omega) = X(\varphi^n \omega)$ then we obtain a stochastic process on $S$. By the way, is it true that $X$ is always a stationary Markov process?
An event
$ A = \{X_n\in B\text{ infinitely often}\} $ is invariant as well as its complement $ A^c = \{\text{ there exists }N \text{ such that } X_n \in B^c \text{ for all }n\geq N\}. $
Moreover, all invariant events form a $\sigma$-algebra and if $X$ is Markov then its invariant event are characterized by its harmonic functions. However, these characterization are quite elusive, so for me it is difficult to imagine other invariant events based on such characterization.
I would be also grateful if you can refer me to a book/lecture notes where such examples are provided.