Computability of Brjuno-like functions
Dynamical Systems
2026-01-14 v1 Logic
Abstract
In his seminal paper from 1936, Alan Turing introduced the concept of non-computable real numbers and presented examples based on the algorithmically unsolvable Halting problem. We describe a different, analytically natural mechanism for the appearance of non-computability. Namely, we show that additive sampling of orbits of certain skew products over expanding dynamics produces Turing non-computable reals. We apply this framework to Brjuno-type functions to demonstrate that they realize bijections between computable and lower-computable numbers, generalizing previous results of M. Braverman and the second author for the Yoccoz-Brjuno function to a wide class of examples, including Wilton's functions and generalized Brjuno functions.
Cite
@article{arxiv.2501.04195,
title = {Computability of Brjuno-like functions},
author = {Ivan O. Shevchenko and Michael Yampolsky},
journal= {arXiv preprint arXiv:2501.04195},
year = {2026}
}