10 problems

Let $T$ be a rank-one transformation associated with a sequence $(r_n,s_n)_{n=1}^\infty$, where the classical staircase case is $r_n=n$. Smorodinsky's staircase mixing conjecture.…

A rank-one map is a rank-one measure-preserving transformation, and its maximal spectral type is the spectral measure class arising from the associated Koopman operator. Klemes–Rei…

Singular-spectrum conjecture. All rank one maps have singular spectrum; equivalently, every rank one transformation has singular maximal spectral type.

Nonintegrability conjecture. The logarithm of the absolutely continuous part of the spectrum is not integrable.

Let $T$ be a rank-one automorphism, let $U_T$ be its Koopman operator, and let $J_2^e(T)$ denote the set of ergodic self-joinings. For each $\rho\in J_2^e(T)$, write $J=J_\rho$ for…

Conjecture. There exists an ergodic map $\tau$ of $X$ such that $f$ is a $\tau$-coboundary with a transfer function in $L_r(X)$. This proposes that, possibly using a rank-one map,…

Borel procedure conjecture. There is a Borel procedure for determining whether two rank-one transformations allow eventually commensurate cutting and spacer parameters.

Thouvenot's conjecture. All bounded mildly mixing constructions have minimal self-joinings.

Let $T$ be a rank one transformation with a fixed sequence of tower partitions $(\xi_n)$ such that $\xi_{n+1}$ refines $\xi_n$ and every measurable set $A$ can be approximated by a…

Power weak mixing conjecture. Every high staircase transformation is power weakly mixing.