Existence of a "universal" measure-preserving transformation on the unit interval

Question

Let $I = [0,1]$ be the unit interval equipped with the Lebesgue measure $\lambda$. Let $\mathcal{M}$ be the set of all Lebesgue measure-preserving transformations $T: I \to I$. We say a transformation $T \in \mathcal{M}$ is universal if for any other transformation $S \in \mathcal{M}$, there exists a measurable subset $A \subseteq I$ with $\lambda(A) > 0$ such that the restriction of $T$ to $A$, denoted $T|_A$, is measure-theoretically isomorphic to $S$.

Does a universal measure-preserving transformation exist on the unit interval?

Answer 1

As written, this has no chance. If $A$ isn't invariant, what does it mean to say $T|_A$ is isomorphic to $S$? Even if you restrict to invariant sets, you still have a problem with the positive measure condition: for each $0<\alpha<1$, consider the Lebesgue measure-preserving transformation $T_\alpha(x)=x/\alpha$ if $\in [0,\alpha)$ and $T_\alpha(x)=(x-\alpha)/(1-\alpha)$ if $x\in [\alpha,1)$. These are ergodic and have distinct entropies. If your condition were to hold, there would be a set $A_\alpha$ of positive measure for each $T_\alpha$. Since each $T_\alpha$ is ergodic, the subset $A_\alpha$ would have no proper invariant subsets of positive measure. It follows that the $A_\alpha$ are disjoint, which is impossible.

If you weaken your definition of universality to saying there is a subset (with no assumptions about the measure) on which $T$ behaves like your favourite map, this is (almost certainly) true. An old paper of Weiss contains essentially this result (with a different space than $[0,1]$ and a stronger condition on $T$).

B. Weiss, Countable generators in dynamics—universal minimal models. Measure and measurable dynamics (Rochester, NY, 1987), 321–326. Contemp. Math., 94 American Mathematical Society, Providence, RI, 1989