What is an unique ergodic measure?

Question

In a book I found the following:

[...] on spaces of the form $\Omega\times \mathbb {R}^2$, where $\Omega $ carries a unique ergodic measure.

What is meant with $\Omega$ carries a unique ergodic measure?

I only know what ergodic transformations are.

Answer 1

There are transformations which have a lot of invariant measures (some of them ergodic, others not), but there are also transformations which admit a unique invariant measure which turns out to be ergodic. In the latter case the transformation is called uniquely ergodic.

For example, consider a rotation of the circle (one of the most basic standard transformations in ergodic theory). Here the circle will be denoted as $\mathbb R/\mathbb Z$ and the transformation is $R_\alpha(x)=x+\alpha \mod 1$. There are two cases:

$\bullet$ The rotation is rational, i.e. $\alpha$ is a rational number, let's say $\alpha=p/q$ in reduced form. In this case every point in the circle belongs to a (finite) periodic orbit of length $q$ and every one of those orbits can be the support of an $R_\alpha$-invariant measure, i.e. the uniform measure giving mass $1/q$ to each point in the single periodic orbit. (It is easy to check that this measure is ergodic and in fact all ergodic measures of the rational rotation are of this form.) Nevertheless there are many more invariant measures, in particular combinations of uniform measures in various periodic orbits. (In fact the ergodic decomposition theorem states that in this case every $R_\alpha$-invariant measure can be written as a "sum", better an integral of ergodic measures which are uniform measures in single orbits.)

$\bullet$ The second case is an irrational rotation, where $\alpha$ is an irrational number. In this case it can be shown that every orbit is in fact dense in the circle and that there is a unique (rotation-invariant) measure, namely the Lebesgue or Haar measure of the circle. Again this measure is ergodic and thus the transformation $R_\alpha$ with $\alpha$ irrational is uniquely ergodic.

In your case, to say $\Omega$ carries a unique ergodic measure, probably means that there is already a transformation on $\Omega$ and with respect to this transformation there is a unique invariant measure (which is ergodic for this transformation).