Matthijs J. Borst | Searcharxiv

arXiv2021

A multidimensional solution to additive homological equations

In this paper we prove that for a finite-dimensional real normed space $V$, every bounded mean zero function $f\in L_\infty([0,1];V)$ can be written in the form $f = g\circ T - g$ for some $g\in L_\infty([0,1];V)$ and some ergodic invertible measure preserving transformation $T$ of $[0,1]$. Our method moreover allows us to choose $g$, for any given $\varepsilon>0$, to be such that $\|g\|_\infty\leq (S_V+\varepsilon)\|f\|_\infty$, where $S_V$ is the Steinitz constant corresponding to $V$.

math.DS↗

arXiv2019

Full proof of Kwapień's theorem on representing bounded mean zero functions on $[0,1]$

In [7], Kwapień announced that every mean zero function $f\in L_\infty[0,1]$ can be written as a coboundary $f = g\circ T -g$ for some $g\in L_\infty[0,1]$ and some measure preserving transformation $T$ of $[0,1]$. Whereas the original proof in [7] holds for continuous functions, there is a serious gap in the proof for functions with discontinuities. In this article we fill in this gap and establish Kwapień's result in full generality. Our method also allows to improve the original result by showing that for any given $ε>0$ the function $g$ can be chosen to satisfy a bound $\|g\|_\infty\leq (1+ε)\|f\|_\infty$.

math.DS↗