2024/06/20 by Dario Corona, Alessandro Della Corte, Corona, Dario +3
Physics and Astronomy · #37B20 #37B40 #Dynamical Systems (math.DS) #FOS: Mathematics #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2406.14487
openalex publication_date 2024/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Mathématique, 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 Mathématique, 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 Mathématique, 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.