# Does an “almost mixing” transformation admit a non-null ergodic component?

Problem set up:

Let $$\mathbf X := (X, \mathcal A, \mu)$$ be a standard probability space.

We say that a measure preserving transformation $$T$$ on $$\mathbf X$$ is $$\varepsilon$$-almost mixing if for every $$\delta > \varepsilon$$, and every pair of non-null measurable sets $$A, B \in \mathcal A$$, there exists an $$N > 0$$ such that for all $$n > N$$, we have $$|\mu(T^{-n}A \cap B) - \mu(A)\mu(B)| < \delta \mu(A)\mu(B)$$.

We say a measure preserving transformation $$G$$ on $$X$$ admits an ergodic component if there exists some non-null measurable subset $$E$$ of $$X$$ such that $$G(E) \subset E$$, and the “restricted system” ($$\mathbf E, G_{|E})$$, with $$\mathbf E := (E, \mathcal A_{|E}, \mu_{|E})$$ is ergodic. Here $$\mathcal A_{|E}$$ is the restricted sigma algebra, and $$\mu_{|E}$$ is defined by $$\mu_{|E}(A) := \mu(A \cap E)/\mu(E)$$.

Question: Does there exist some $$\varepsilon > 0$$ such that any $$\varepsilon$$-almost mixing transformation $$T$$ on $$\mathbf X$$ admits an ergodic component?

A paper of Martin and England (https://www.ams.org/journals/bull/1968-74-03/S0002-9904-1968-11982-2/S0002-9904-1968-11982-2.pdf) shows (in your language) that if $$T$$ is $$\epsilon$$-almost mixing for any $$\epsilon<1$$, then $$T$$ is weak-mixing (and hence ergodic).
• Actually the argument is quite simple from the von Neumann characterization of weak mixing. If the dynamical system is not weak mixing, it had a rotation factor. In that factor, you can find sets $A$ and $B$ such that $\mu(T^{-n}A\cap B)=0$ for infinitely many $n$. These sets can be lifted to the full system. This shows the full system is not almost mixing. Hence almost mixing implies weak mixing. May 14 at 15:31