DSNT Seminar: The Brjuno and the 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.

I will discuss how these two functions are related to each other also by the means of a generalization of the Brjuno function - the so-called semi-Brjuno function B_0(x) - built using the by-excess continued fraction map. Namely, B(x) and W (x) can be expressed in terms of the even B_0^+ and odd B_0^- parts of B_0(x), respectively, up to a bounded defect. Moreover we study the regularity of the differences

∆+(x) = B+(x) − 2B_0^+ (x) and ∆−(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.

This is joint work with Claire Burrin (University of Zurich) and Seul Bee Lee (Seoul National University): see https://arxiv.org/abs/2503.08206

Bibliography:

• [Brj65] A. D. Brjuno, On convergence of transforms of differential equations to the normal form (Russian), Dokl. Akad. Nauk SSSR (Russian Academy of Sciences) 165 (1965), 987–989.
• [LMPS24] Lee, S.B., Marmi, S., Petrykiewicz, I., Schindler, T.I.,, Regularity properties of k-Brjuno and Wilton functions. Aequat. Math. 98, 13–85 (2024).
• [MMY06] Marmi, S., Moussa, P. and Yoccoz, J.C., 2006. Some properties of real and complex Brjuno functions. Frontiers in Number Theory, Physics, and Geometry I, pp.603-625.
• [Wil33] J.R. Wilton, An approximate functional equation with applications to a problem of Diophantine approximation. Journal für die reine und angewandte Mathematik 169 (1933) 219-237
• [Yoc95] J.C. Yoccoz, Théorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque, 231 (1995), 3–88.

This is a seminar in our seminar series on Dynamical Systems and Number Theory (DSNT).