A solution to the multidimensional additive homological equation

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Abstract

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 $V$.

About the authors

Aleksei Feliksovich Ber

National University of Uzbekistan named after M. Ulugbek, Faculty of Mathematics and Mechanics

Email: ber@ucd.uz
Candidate of physico-mathematical sciences, Senior Researcher

Matthijs Borst

Delft University of Technology

Email: m.j.borst@outlook.com

Sander Borst

Centrum voor Wiskunde en Informatica

Email: sander.borst@cwi.nl

Fedor Anatol'evich Sukochev

University of New South Wales, School of Mathematics and Statistics

Email: f.sukochev@unsw.edu.au
Candidate of physico-mathematical sciences, Professor

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