Existence and non-existence of solutions to the coboundary equation for measure-preserving systems

Abstract A fundamental question in the field of cohomology dynamical systems is to determine when there are solutions coboundary equation: $$ \begin{align*} f = g - \circ T. \end{align*} In many cases, T given be an ergodic invertible measure-preserving transformation on a standard probability space $(X, {\mathcal B}, \mu )$ and contained $L^p$ for $p \geq 0$ . We extend previous results by showing any measurable that non-zero set positive measure, class with solution meager (including case where $\int _X f\,d\mu ). From this fact, natural arises: , does always exist pair ? regards question, our main as follows. Given function such $f(x) g(x) g(Tx)$ almost every (a.e.) $x\in X$ if only _{f> 0} \int _{f < $ (whether finite or $\infty mean-zero $f \in L^p(\mu 1$ $g L^{p-1}(\mu g( Tx a.e. $x some sense, existence result best possible. For exists dense $G_{\delta }$ everywhere, then \notin L^q(\mu $q> p Finally, it shown we cannot expect moments L^1(\mu particular, $\lim _{x\to \infty } \phi (x) transfer satisfies: \int_{X} ( | )\,d\mu \infty.