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Piecewise contracting maps on the interval: Hausdorff dimension, entropy, and attractors

Part of: Topological dynamics Low-dimensional dynamical systems

Published online by Cambridge University Press:  21 September 2023

Alfredo E. Calderón Open the ORCID record for Alfredo E. Calderón [Opens in a new window]  and
Edgardo Villar-Sepúlveda Open the ORCID record for Edgardo Villar-Sepúlveda [Opens in a new window]
Alfredo E. Calderón*
Affiliation:
Escuela de Ingeniería, Facultad de Ingeniería y Empresa, Universidad Católica Silva Henríquez, Gral. Jofré #462, Santiago, Región Metropolitana 8330226, Chile
Edgardo Villar-Sepúlveda
Affiliation:
Department of Engineering Mathematics, University of Bristol, Ada Lovelace Building, Tankard’s Cl, University Walk, Bristol BS8 1TW, UK e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

We consider the attractor $\Lambda $ of a piecewise contracting map f defined on a compact interval. If f is injective, we show that it is possible to estimate the topological entropy of f (according to Bowen’s formula) and the Hausdorff dimension of $\Lambda $ via the complexity associated with the orbits of the system. Specifically, we prove that both numbers are zero.

Keywords

Interval mappiecewise contractionattractorcomplexityHausdorff dimension

MSC classification

Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 37B35: Gradient-like and recurrent behavior; isolated (locally maximal) invariant sets 37B10: Symbolic dynamics
Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 2 , June 2024 , pp. 318 - 327
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

A.E.C. was supported by ANID Fondecyt Iniciación N°11230064, ANID Fondecyt Regular N°1230569, MathAmsud Project VOS 22-MATH-08, and MathAmsud Project TOMCAT 22-MATH-10. E.V-S. was supported by Ph.D. funding from ANID, Beca Chile Doctorado en el Extranjero (Grant No. 72210071).

References

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