DOI: 10.4213/im9319e ISSN:

A solution to the multidimensional additive homological equation

Aleksei Feliksovich Ber, Matthijs Borst, Sander Borst, Fedor Anatol'evich Sukochev
  • General Mathematics

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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