Dynamical properties of critical exponent functions

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Detalles Bibliográficos
Autores principales:	Corona, Dario, Della Corte, Alessandro, Farotti, Marco
Formato:	Preprint
Publicado:	2024
Materias:	Dynamical Systems 37B40, 37B20
Acceso en línea:	Acceder al recurso
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_version_	1866909273432260608
author	Corona, Dario Della Corte, Alessandro Farotti, Marco
author_facet	Corona, Dario Della Corte, Alessandro Farotti, Marco
contents	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.
format	Preprint
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institution	arXiv
publishDate	2024
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spellingShingle	Dynamical properties of critical exponent functions Corona, Dario Della Corte, Alessandro Farotti, Marco Dynamical Systems 37B40, 37B20 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.
title	Dynamical properties of critical exponent functions
topic	Dynamical Systems 37B40, 37B20
url	https://arxiv.org/abs/2406.14487

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