Dyadic potential theory and de Rham functions

Abstract

We study de Rham functional equations driven by two increasing fractional linear transformations. Our main purpose is to relate the singularity theory of the associated solutions to dyadic potential theory on the binary tree. We first prove an existence and uniqueness theorem for increasing, left-continuous solutions in the full range of linear fractional data, and identify the trapping region in parameter space where the solution is continuous. For a large class of parameters we show that the de Rham solution is the normalized cumulative capacitary function of a multiplicative dyadic capacity. This gives a potential-theoretic model for Möbius de Rham systems. We then sharpen Okamura's Hausdorff-dimensional estimates for the singular measure associated with the solution by replacing Hausdorff dimension with dyadic Riesz capacities at the upper endpoint of Okamura's theorem.

Disclosure

“yadic capacities satisfy recursive relations which, under some self-similarity assumptions on the involved weights, lead to a multiplicity of de Rham systems. We will return on some of these in a subsequent article. Methodological note. A generative AI assistant was used during the preparation of this manuscript for routine algebraic manipulations, numerical and symbolic checks, bibliographic and terminological queries, and for drafting preliminary versions of some arguments from proof s”

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2026-08-24T00:34:15.228Z

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