Computability of Brjuno-like functions

TL;DR

This paper explores the non-computability of certain mathematical functions, showing that specific dynamical systems can produce non-computable reals and extending previous results on Brjuno functions to a broader class.

Contribution

It introduces a new analytical mechanism for non-computability via skew product dynamics and generalizes prior work on Brjuno functions to a wider class of examples.

Findings

Additive sampling of skew product orbits produces non-computable reals.

Brjuno-type functions can bijectively map computable to lower-computable numbers.

Generalization of previous results to Wilton's and other generalized Brjuno functions.

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.