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

Abstract

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