A 0–1 law for multiplicative functional equations

Abstract

The 0–1 principle stating that for the multiplicative functional equation $$\prod_{i=1}^m f_i(x_i)=\prod_{j=1}^n g_j(y_j)$$ satisfied almost everywhere one of the unknown functions \({f_{i}}\)’s and \({g_{j}}\)’s is zero almost everywhere on both sides or all of them are nonzero almost everywhere is generalized for functions defined on connected manifolds \({X_{i}}\)’s and \({Y_{j}}\)’s (the \({y_{j}}\)’s are given functions). Corollaries proving that the unknown functions are almost equal to nowhere zero \({\mathcal{C}^{\infty}}\) functions satisfying the functional equation everywhere are also given. The Baire category analogues is also treated.