The Brjuno and Wilton Functions

Abstract

The Brjuno and Wilton functions bear a striking resemblance, despite their very different origins; while the Brjuno function $B(x)$ is a fundamental tool in one-dimensional holomorphic dynamics, the Wilton function $W(x)$ stems from the study of divisor sums and self-correlation functions in analytic number theory. We show that these perspectives are unified by the semi-Brjuno function $B_0(x)$. Namely, $B(x)$ and $W(x)$ can be expressed in terms of the even and odd parts of $B_0(x)$, respectively, up to a bounded defect. Based on numerical observations, we further analyze the arising functions $\hat{I}{”}^{+}(x)=B^{+}(x)-2B_0^{+}(x)$ and $\hat{I}{”}^{-}(x)=W^{-}(x)-2B_0^{-}(x)$, the first of which is Hölder continuous whereas the second exhibits discontinuities at rationals, behaving similarly to the classical popcorn function.

10 pages, 3 figures