ergodicity of the group of transformations preserving a partition - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:13:12Z http://mathoverflow.net/feeds/question/96068 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96068/ergodicity-of-the-group-of-transformations-preserving-a-partition ergodicity of the group of transformations preserving a partition Stéphane Laurent 2012-05-05T14:19:43Z 2012-05-05T18:19:54Z <p>Let $X=\{0,1\}^{\mathbb{N}}$ and $\theta$ be the partition of $X$ induced by the equivalence relation $x \sim x'$ when $x$ and $x'$ differ only at a finite number of coordinates (see <a href="http://mathoverflow.net/questions/96023/intersection-partition-as-an-orbital-partition" rel="nofollow">this related question</a>).</p> <p>Given a Bernoulli measure $m$ on $X$, let ${\cal H}$ be the group of transformations $S$ of $X$ satisfying $\theta(x)=\theta(S(x))$ for almost all $x$ and let ${\cal G}$ be the subgroup of ${\cal H}$ consisting of measure-preserving transformations.</p> <p>Is it possible to explicitely describe ${\cal G}$ ? Under which condition on $m$ the group ${\cal G}$ is ergodic ?</p> <p>EDIT: I am also interested in the case when $m$ is a stationnary Markov probability on $X$.</p> http://mathoverflow.net/questions/96068/ergodicity-of-the-group-of-transformations-preserving-a-partition/96075#96075 Answer by Jesse Peterson for ergodicity of the group of transformations preserving a partition Jesse Peterson 2012-05-05T16:21:51Z 2012-05-05T18:19:54Z <p>It's easy to see that the action is always ergodic since $\mathcal G$ contains the group of finite permutations on the indices, which acts ergodically. In fact, the group $\mathcal G$ (which equals $\mathcal H$ in the case $m = \mu^{\mathbb N}$ with $\mu(0) = 1/2$.) that you are describing is the full group of the ergodic hyperfinite measurable equivalence relation. It, and other full groups are discussed in Sections I.3 and I.4 in the book by Alexander Kechris: Global aspects of ergodic group actions, Mathematical Surveys and Monographs, 160, American Mathematical Society, 2010. </p>