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 $X$.

  • $\begingroup$ 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

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