A converse result concerning the periodic structure of commuting affine circle maps

Volume 9, Issue 7, pp 5041--5060 http://dx.doi.org/10.22436/jnsa.009.07.08

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Authors

José Salvador Cánovas Peña - Departamento de Matematica Aplicada y Estadistica, Universidad Politecnica de Cartagena, Campus Muralla del Mar, 30203{Cartagena, Spain. Antonio Linero Bas - Department of Mathematics, Universidad de Murcia, Campus de Espinardo, 30100-Murcia, Spain. Gabriel Soler López - Departamento de Matematica Aplicada y Estadistica, Universidad Politecnica de Cartagena, Alfonso XIII 52, 30203{Cartagena, Spain.

Abstract

We analyze the set of periods of a class of maps \(\phi_{d,\kappa}: \mathbb{Z}_\Delta\rightarrow \mathbb{Z}_\Delta\) defined by \(\phi_{d,\kappa}(x)=dx+\kappa,\quad d,\kappa\in\mathbb{Z}_\Delta\), where \(\Delta\) is an integer greater than 1. This study is important to characterize completely the period sets of alternated systems \(f; g; f; g,... \), where \(f; g : \mathbb{S}_1 \rightarrow \mathbb{S}_1\) are affine circle maps that commute, and to solve the converse problem of constructing commuting affine circle maps having a prescribed set of periods.

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Keywords

• Affine maps
• alternated system
• periods
• circle maps
• degree
• combinatorial dynamics
• ring of residues modulo m
• Abelian multiplicative group of residues modulo m
• Euler function
• congruence
• order
• generator.

MSC

•  37E10
•  11A07

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