Unique Ergodic Measure

Question

From <http://mathworld.wolfram.com/ErgodicMeasure.html>

"If there is only one ergodic measure, then $T$ is called uniquely ergodic. An example of a uniquely ergodic transformation is the map $x\to x+a\mod 1$ on the unit interval when $a$ is irrational. Here, the unique ergodic measure is Lebesgue measure."

1. If we start with a point and consider its orbit with respect to the map $T$, what is the Lebesgue measure on?
2. Can someone give an example of a system which has more than one ergodic measure?

Answer 1

Im not sure what your first question means, but in regards the second:

If $\{1,2,3,...,n\}^{\mathbb{Z}}$ is given the infinite product measure coming from a probability vector $(a_1,...a_n)$ $ ( \sum_ia_i=1)$ then the Bernoulli shifts (just shift every entry to the left one space) are always ergodic. Since there are different choices for the probability vector, there are diffent ergodic measures for the Bernoulli shift. See chapter two of Einseidler and Wards book "Ergodic theory with a view toward number theory".