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

Question

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?

Answer 1

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).