A multidimensional solution to additive homological equations

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 $\epsilon >0$, to be such that $\Vert g{\Vert }_{\infty }\le (S_V+\epsilon )\Vert f{\Vert }_{\infty }$, where $S_V$ is the Steinitz constant corresponding to $V$.

51 pages