3
$\begingroup$

Here is a lemma that I know to be true, and can prove in half a page or so, but I'm wondering: can anyone supply a reference so that it can simply be quoted in a paper?

Lemma Let $T$ be an ergodic measure-preserving transformation of $(X,\mu)$ and let $n>1$. Then there exists $k$ a factor of $n$ and a set $B$ of measure $1/k$ such that $T^n|_B$ is ergodic. The sets $(T^{-j}B)_{j=0}^{k-1}$ partition $X$.

$\endgroup$
  • $\begingroup$ Are you asking about the ergodic components of $T^n$? $\endgroup$ – R W Nov 18 '18 at 15:21
  • $\begingroup$ @RW : yes. Exactly. $\endgroup$ – Anthony Quas Nov 18 '18 at 19:26
  • $\begingroup$ It must be mentioned somewhere as an exercise :) The space of ergodic components of $T^n$ is naturally endowed with a measure preserving action of the finite cyclic group $\mathbb Z_n$, which is ergodic by the ergodicity of $T$. Now, any ergodic measure preserving action of $\mathbb Z_n$ is a cyclic shift on $k$ points; the preimage of any of these points is your set $B$. $\endgroup$ – R W Nov 18 '18 at 21:26
  • $\begingroup$ I agree. This should be an exercise somewhere. But I haven’t been able to locate it. My briefest proof so far says: find the largest factor k of n such that $U_T$ has an eigenvalue $e^{2\pi I/k}$. Then $B$ is a level set of the eigenfunction. $\endgroup$ – Anthony Quas Nov 18 '18 at 22:24
  • $\begingroup$ Tastes differ :) $\endgroup$ – R W Nov 18 '18 at 23:21

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.