Dynamical properties of critical exponent functions

TL;DR

This paper investigates the topological dynamical properties of critical exponent functions, identifies an error in a previous proof, and recovers most results while proposing a weaker conjecture for the corrected theorem.

Contribution

The paper corrects an error in a key combinatorial theorem about critical exponent maps and offers a revised conjecture, advancing understanding of their chaotic properties.

Findings

Most previous results are recoverable despite the error.

A weaker form of the original theorem is proposed as a conjecture.

The study enhances understanding of the chaotic behavior of critical exponent maps.

Abstract

In the last years the attention towards topological dynamical properties of highly discontinuous maps has increased significantly. In [D.Corona, A. Della Corte. The critical exponent functions. Comptes Rendus Math\'ematique, 360(G4), 315-332, 2022], a class of densely discontinuous interval maps, called "critical exponent maps", was introduced. These maps are defined through the word-combinatorics concept of critical exponent applied to the binary expansion of reals and show highly chaotically properties as well as some challenging problems. In this paper we identify an error in the proof of Theorem 7 in [D.Corona, A. Della Corte. The critical exponent functions. Comptes Rendus Math\'ematique, 360(G4), 315-332, 2022], a purely combinatorial result which in fact does not hold. We show that most of the results in [D.Corona, A. Della Corte. The critical exponent functions. Comptes Rendus Math\'ematique, 360(G4), 315-332, 2022], obtained there through Theorem 7, can be recovered. Moreover, we propose as a conjecture a weaker form of Theorem 7.