A converse result concerning the periodic structure of commuting affine circle maps
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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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ISRP Style
José Salvador Cánovas Peña, Antonio Linero Bas, Gabriel Soler López, A converse result concerning the periodic structure of commuting affine circle maps, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 7, 5041--5060
AMA Style
Peña José Salvador Cánovas, Bas Antonio Linero, López Gabriel Soler, A converse result concerning the periodic structure of commuting affine circle maps. J. Nonlinear Sci. Appl. (2016); 9(7):5041--5060
Chicago/Turabian Style
Peña, José Salvador Cánovas, Bas, Antonio Linero, López, Gabriel Soler. "A converse result concerning the periodic structure of commuting affine circle maps." Journal of Nonlinear Sciences and Applications, 9, no. 7 (2016): 5041--5060
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
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