Sinai's factor theorem

Remarks

A factor map can only lower entropy, so the hypothesis $h(B) \le h_\mu(T)$ is necessary. Ergodicity is needed too. Let $T$ be the disjoint union, with weight $1/2$ each, of a Bernoulli scheme $B'$ and an identity map, so that $h_\mu(T) = h(B')/2 > 0$. Let $B$ be any Bernoulli scheme with at least two symbols of positive weight. A factor map $\varphi$ onto $B$ satisfies $\varphi(x) = B(\varphi(x))$ on the identity part. So it sends a set of measure $1/2$ into the fixed points of $B$, which form a null set. So $T$ has no such factor, whatever the entropy of $B$. The proof constructs the factor map as a limit of finite codings, using the Rokhlin lemma to build towers and the Shannon–McMillan–Breiman theorem to count typical names. Example: an irrational rotation has entropy $0$, so the theorem says nothing about it. Indeed it has no nontrivial Bernoulli factor, since every factor of a zero-entropy system has entropy $0$. Example: every ergodic toral automorphism of positive entropy has Bernoulli factors of every entropy up to its own.

Going deeper

For a student

It was a step toward the Ornstein isomorphism theorem, whose proof sharpens the same kind of coding argument from factor maps to isomorphisms. It shows that positive entropy, a number defined by counting, forces a concrete structure: an independent process sitting inside the system.