# Powers of ergodic transformations

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 $$X$$.

• Are you asking about the ergodic components of $T^n$? – R W Nov 18 '18 at 15:21
• @RW : yes. Exactly. – Anthony Quas Nov 18 '18 at 19:26
• 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 $B$. – R W Nov 18 '18 at 21:26
• 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. – Anthony Quas Nov 18 '18 at 22:24
• Tastes differ :) – R W Nov 18 '18 at 23:21