I was recently reading some textbooks and topics in Ergodic theory that I found this fascinating result which I couldn't prove completely:
Suppose that $T$ and $S$ are two Ergodic transformations. Show that $T*S$ is Ergodic if and only if the only common eigenvalue of both $T$ and $S$ is $1$.
In the above, $T*S$ denotes the product of the two transformations.