Entropy and recurrent dimensions of discrete dynamical systems given by p-adic expansions

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• 作者：H. Inoue ; K. Naito
• 关键词：p ; adic theory ; symbolic dynamics ; recurrences
• 刊名：P-Adic Numbers, Ultrametric Analysis, and Applications
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：7
• 期：2
• 页码：157-167
• 全文大小：527 KB
• 参考文献：1.B. Adamczewski and Y. Bugeaud, 鈥淥n the complexity of algebraic numbers I. Expansions in integer bases,鈥?Annals Math. 165, 547鈥?65 (2007).View Article MATH MathSciNet
  2.M. Morse and G. A. Hedlund, 鈥淪ymbolic dynamics II-Sturmian trajectories,鈥?Amer. J. Math. 62, 1鈥?2 (1940).View Article MathSciNet
  3.K. Naito, 鈥淒imension estimate of almost periodic attractors by simultaneous Diophantine approximation,鈥?J. Diff. Equations 141, 179鈥?00 (1997).View Article MATH MathSciNet
  4.K. Naito, 鈥淩ecurrent dimensions of quasi-periodic solutions for nonlinear evolution equations,鈥?Trans. Amer. Math. Soc. 354(3), 1137鈥?151 (2002).View Article MATH MathSciNet
  5.K. Naito, 鈥淩ecurrent dimensions of quasi-periodic solutions for nonlinear evolution equations II: Gaps of dimensions and Diophantine conditions,鈥?Discrete Cont. Dyn. Syst. 11, 449鈥?88 (2004).View Article MATH MathSciNet
  6.K. Naito, 鈥淐lassifications of irrational numbers and recurrent dimensions of quasi-periodic orbits,鈥?J. Nonl. Anal. Convex Anal. 5, 169鈥?85 (2004).MATH MathSciNet
  
• 作者单位：H. Inoue (1)
  K. Naito (1)
  
  1. Department of Applied Mathematics, Graduate School of Science and Technology Kumamoto University, Kurokami 2-39-1, Kumamoto, Japan
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algebra
  Russian Library of Science
  
• 出版者：MAIK Nauka/Interperiodica distributed exclusively by Springer Science+Business Media LLC.
• ISSN：2070-0474

文摘

In this paper we study symbolic dynamical systems given by the shift mapping on the coefficient sequences of expansions of p-adic numbers. We associate the upper and lower recurrent dimensions with the topological entropies of these discrete dynamical systems by giving some inequalities representing the relationships among these parameters. Using these inequality relations, we estimate the topological entropies or recurrent dimensions of the various coefficient sequences, which have recurrent properties. For the case of Sturmian sequences we can estimate the positive gap values between upper and lower recurrent dimensions, which indicate the unpredictability of the orbits, if the irrational frequencies of the sequences are Liouville numbers.