Existence and Non-existence of Solutions to the Coboundary Equation for Measure Preserving Systems

Let $(X,mathcal{B},mu)$ be a standard probability space. We give new fundamental results determining solutions to the coboundary equation: begin{eqnarray*} f = g - g circ T end{eqnarray*} where $f in L^p$ and $T$ is ergodic invertible measure preserving on $(X, mathcal{B}, mu )$. 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 $int_X f dmu = 0$). 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: (i) Given measurable $f$, there exists an ergodic invertible measure preserving transformation $T$ and measurable function $g$ such that $f(x) = g(x) - g(Tx)$ for a.e. $xin X$, if and only if $int_{f > 0} f dmu = - int_{f < 0} f dmu$ (whether finite or $infty$). (ii) Given mean-zero $f in L^p$ for $p geq 1$, there exists an ergodic invertible measure preserving $T$ and $g in L^{p-1}$ such that $f(x) = g(x) - g( Tx )$ for a.e. $x in X$. (iii) In some sense, the previous existence result is the best possible. For $p geq 1$, there exist mean-zero $f in L^p$ such that for any ergodic invertible measure preserving $T$ and any measurable $g$ such that $f(x) = g(x) - g(Tx)$ a.e., then $g otin L^q$ for $q > p - 1$. Also, we show this situation is generic for mean-zero $f in L^p$. Finally, it is shown that we cannot expect finite moments for solutions $g$, when $f in L^1$. In particular, given any $phi : mathbb{R} to mathbb{R}$ such that $lim_{xto infty} phi (x) = infty$, there exist mean-zero $f in L^1$ such that for any solutions $T$ and $g$, the transfer function $g$ satisfies: begin{eqnarray*} int_{X} phi big( | g(x) | big) dmu = infty. end{eqnarray*}