A multidimensional solution to additive homological equations

TL;DR

This paper demonstrates that in finite-dimensional normed spaces, any bounded mean zero function can be decomposed into a coboundary with controlled norm, using ergodic measure-preserving transformations, extending solutions to additive homological equations.

Contribution

It provides a multidimensional approach to solving additive homological equations with norm control in finite-dimensional spaces.

Findings

Any bounded mean zero function can be expressed as a coboundary with an ergodic transformation.

The method allows for norm control of the transfer function g, related to the Steinitz constant.

The approach extends classical results to a multidimensional setting with explicit bounds.

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 $V$.