Dyadic potential theory and de Rham functions

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.

In particular, we prove capacitary upper and lower bounds for the size of Borel sets carrying full or zero mass, and obtain a capacitary sharpening of Okamura's singularity criterion.