Square roots of measure-preserving automorphisms

Characterize those Lebesgue-measure automorphisms T of the unit interval that have a measure-preserving automorphism S with T=S-squared.

Background

The problem is attributed to Halmos and asks which measure-preserving transformations admit square roots. The paper additionally records a conjecture that the set of such transformations is Sigma-one-one-complete.

References

What $T \in \text{Aut}([0,1],\lambda)$ has $S \in \text{Aut}([0,1],\lambda)$ with $T = S2 = S \circ S$?

Conjecture: ${T \in \text{Aut}([0,1],\lambda) : \exists S \in \text{Aut}([0,1],\lambda) \, T = S2}$ is $\mathbf{\Sigma}_11$-complete.

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in 2025 Section 14