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arXiv · 2107.11248

A multidimensional solution to additive homological equations

Aleksei F. Ber·Matthijs J. Borst·Sander J. Borst·Fedor A. Sukochev
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Abstract

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 $V$.

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Aleksei F. Ber, Matthijs J. Borst, Sander J. Borst, Fedor A. Sukochev. 2021-07-23. A multidimensional solution to additive homological equations. https://arxiv.org/abs/2107.11248

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