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A 0–1 law for multiplicative functional equations

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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.

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Correspondence to Antal Járai.

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Dedicated to Professor Roman Ger to his 70th birthday

This work has been supported by OTKA K111651 grant.

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Járai, A. A 0–1 law for multiplicative functional equations. Aequat. Math. 90, 147–161 (2016). https://doi.org/10.1007/s00010-016-0407-1

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  • DOI: https://doi.org/10.1007/s00010-016-0407-1

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