3 problems

Let $f\in L^p_0$ be a specified function. An ergodic invertible measure-preserving transformation is a transformation $T$ preserving the measure and having no nontrivial invariant…

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,…

Conjecture. For every such $\phi$, there exists an explicitly constructible mean-zero function $f\in L_\infty(X)$ such that