Rokhlin lemma

Remarks

The sets $B, TB, \dots, T^{n-1}B$ form a Rokhlin tower of height $n$ with base $B$, and $T$ moves each level to the next one. The proof for ergodic $T$ starts from a set $F$ of small positive measure and slices the space by the first return time to $F$. Each column of height at least $n$ is cut into blocks of height $n$, and the leftover has measure at most about $n \mu(F)$. Example at work: a tower gives a periodic approximation of $T$. Define $P = T$ on $B \cup \cdots \cup T^{n-2}B$, let $P = T^{-(n-1)}$ on $T^{n-1}B$, and let $P$ be the identity elsewhere. Then $P^n$ is the identity, $P$ preserves $\mu$, and $P$ agrees with $T$ outside a set of measure less than $\mu(B) + \varepsilon \le 1/n + \varepsilon$. The aperiodicity hypothesis is necessary: if $T^2 = \mathrm{id}$ and $n = 3$, then $T^2 B = B$, so condition 1 forces $\mu(B) = 0$ and condition 2 fails. The error $\varepsilon$ cannot be removed in general. If a tower of height $n \ge 2$ covers $X$ up to a null set, then $T^n B = B$ up to a null set, and $f = \sum_{j=0}^{n-1} e^{2\pi i j/n} \mathbf{1}_{T^j B}$ satisfies $f \circ T = e^{2\pi i/n} f$. So $f$ is a nonconstant eigenfunction, which is impossible when $T$ is weakly mixing, for example a Bernoulli scheme.

Going deeper

For a student

The lemma is the main tool for building approximations and codings in ergodic theory. It shows that every aperiodic transformation can be approximated in the weak topology by periodic ones. It is used in the proofs of Sinai's factor theorem and of Ornstein's isomorphism theorem. It extends to free actions of $\Z^d$ and of countable amenable groups, where the role of intervals of times is played by Følner sets.