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

Volume 9, Issue 7, pp 5041--5060 Publication Date: July 23, 2016
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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.

### 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.

•  37E10
•  11A07

### References

• [1] L. Alsedà, J. Llibre, M. Misiurewicz, Combinatorial dynamics and entropy in dimension one, Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ (1993)

• [2] T. M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg (1976)

• [3] V. I. Arnold, The topology of algebra: combinatorics of squaring, (Russian) Funktsional. Anal. i Prilozhen., 37 (2003), 20-35, translation in Funct. Anal. Appl., 37 (2003), 177--190

• [4] J. S. Cánovas, A. Linero, Periodic structure of alternating continuous interval maps, J. Difference Equ. Appl., 12 (2006), 847--858

• [5] J. S. Cánovas, A. Linero Bas, G. Soler López, Periods of alternated systems generated by affine circle maps, J. Difference Equ. Appl., 22 (2016), 441--467

• [6] A. Linero Bas, Advances in discrete dynamics (Chapter 1. Periodic structure of discrete dynamical systems and global periodicity), Nova Science Publishers, NY, USA (2013)

• [7] O. M. Šarkovskiĭ, Co-existence of cycles of a continuous mapping of the line into itself, (Russian) Ukrain. Mat. Z., 16 (1964), 61--71

• [8] O. M. Šarkovskiĭ, n cycles and the structure of a continuous mapping, (Russian) Ukrain. Mat. Z., 17 (1965), 104--111

• [9] A. N. Šharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, Translated from the Russian by J. Tolosa, Proceedings of the Conference ''Thirty Years after Sharkovskiĭ's Theorem: New Perspectives'', Murcia, (1994), Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 1263--1273

• [10] R. Uribe-Vargas, Topology of dynamical systems in finite groups and number theory, Bull. Sci. Math., 130 (2006), 377--402