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• W2915506842 abstract "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 $int _X f,dmu = 0$ ). 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 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.) $xin X$ , if and only if $int _{f> 0} f,dmu = - int _{f < 0} f,dmu $ (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 _{xto 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: $$ begin{align*} int_{X} phi ( | g(x) | ),dmu = infty. end{align*} $$" @default.
• W2915506842 created "2019-03-02" @default.
• W2915506842 creator A5016567673 @default.
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• W2915506842 date "2022-07-04" @default.
• W2915506842 modified "2023-09-27" @default.
• W2915506842 title "Existence and non-existence of solutions to the coboundary equation for measure-preserving systems" @default.
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• W2915506842 doi "https://doi.org/10.1017/etds.2022.29" @default.
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