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.

27 pages; the statement of theorem 1.3 was corrected and the proof modified accordingly