Why is unique ergodicity important or interesting?

I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be decomposed into smaller subsystems (the only invariant pieces are really big or really small), but once you know that a transformation is ergodic, why do you care if there is only one measure which it's ergodic with respect to or not?

Solution 1:

Unique ergodicity is defined for topological dynamical systems and it tells you that the time average of any function converges pointwise to a constant (see Walters: Introduction to Ergodic Theory, th 6.19). This property is often useful.

Any ergodic measure preserving system is isomorphic to a uniquely ergodic (minimal) topological system (see http://projecteuclid.org/euclid.bsmsp/1200514225).