Abstract
The aim of this paper is to extend the results associated with periodic orbits from two-dimensions to higher-dimensions. Because of the one-to-one correspondence between solutions for the monotone recurrence relation and orbits for the induced high-dimensional cylinder twist map, we consider the system of solutions for monotone recurrence relations instead. By introducing intersections of type (k, l), we propose the definition of generalized Birkhoff solutions, generalizing the concept of Birkhoff solutions. We show that if there is a (p, q)-periodic solution which is not a generalized Birkhoff solution, then the system has positive topological entropy and the Farey interval of p/q is contained in the rotation set.
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Acknowledgements
The author would like to thank Wen-Xin Qin for the helpful discussions and is grateful to the anonymous referee for his/her valuable suggestions.
Funding
National Natural Science Foundation of China, 12201446. Natural Science Foundation of the Jiangsu Higher Education Institutions of China, 22KJB110005. Shuangchuang Program of Jiangsu Province, JSSCBS20220898
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Tong Zhou wrote and reviewed the manuscript.
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Supported by the National Natural Science Foundation of China(12201446), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China(22KJB110005), and the Shuangchuang Program of Jiangsu Province (JSSCBS20220898).
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Zhou, T. Periodic Generalized Birkhoff Solutions and Farey Intervals for Monotone Recurrence Relations. J Dyn Diff Equat (2024). https://doi.org/10.1007/s10884-024-10364-9
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DOI: https://doi.org/10.1007/s10884-024-10364-9
Keywords
- Rotation set
- Generalized Birkhoff solution
- Topological entropy
- Farey interval
- Monotone recurrence relation