# ds.dynamical systems – Does an “almost weak 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 weak 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 $$frac{1}{n} sum_{k = 1}^{n} |mu(T^{-k}A cap B) – mu(A)mu(B)| < delta mu(A)mu(B)$$.

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 weak mixing transformation $$T$$ on $$mathbf X$$ admits an ergodic component?

Remark: This is a potential sharpening of an earlier result.