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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?

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Compare sections 1.6-1.7 to 6.2 in these lecture notes. –  Egbert May 23 '12 at 1:16
Ergodic measures are the extreme points in the convex set of all invariant measures for the particular transformation under consideration. This gives ergodicity an interesting geometric meaning, as the extreme points of a convex set determine the convex set. –  KCd May 23 '12 at 3:17
Let me point out that unique ergodicity is not a property of a measure-preserving transformation: it is meaningless to say that "such-and-such a measure-preserving transformation is uniquely ergodic". Rather, it is a property of a topological dynamical system: you start with a topological dynamical system $(X,f)$, without any invariant measure in the picture at the beginning. Then you ask what invariant measures there can possibly be. Unique ergodicity means there's only one possible invariant measure, and that fact is generally worth knowing. –  Vaughn Climenhaga Aug 29 '12 at 15:59
@VaughnClimenhaga Is it more meaningful to say that "such measure-preserving transformation is isomorphic to a uniquely ergodic transformation" ? –  Stéphane Laurent Mar 18 '13 at 20:33

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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).

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