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.

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techReport A multidimensional solution to additive homological equations A.F. Ber (Aleksei), M.J. Borst (Matthijs), S.J. Borst (Sander) and F.A. Sukochev (Fedor)