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 $V$

A solution to the multidimensional additive homological equation

We prove that, for a finite-dimensional real normed space V, every bounded mean zero function f ∈ L∞([0, 1]; V) can be written in the form f = g ◦ T − g for some g ∈ L∞([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 ε > 0, to be such that ∥g∥∞ ⩽ (SV + ε)∥f∥∞, where SV is the Steinitz constant corresponding to V

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 $V$