Discrete dynamics of one dimensional Collatz like integral value transformations

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• 作者：Sk. Sarif Hassan
• 关键词：Discrete dynamical systems ; Fractal dimension ; Integral value transformations & Collatz like IVTs ; 37D45 ; 37M99 ; 34A30
• 刊名：Journal of Applied Mathematics and Computing
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：49
• 期：1-2
• 页码：91-105
• 全文大小：2,094 KB
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  4.Hassan, Sk. S., Nayak, B. K., Choudhury, P. Pal.: (2012) One Dimensional p-adic Integral Value Transformations, arXiv:鈥?106.鈥?586 (2012) (Under review).
  5.Hassan, Sk. S., Choudhury, P. Pal., Nayak, B. K., Ghosh, A. and Banerjee, J. (2013) Integral Value Transformations: A Class of Affine Discrete Dynamical Systems and an Application, Accepted for Publication.
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• 作者单位：Sk. Sarif Hassan (1)
  
  1. International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore, 560012, India
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Theory of Computation
  Mathematics of Computing
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1865-2085

文摘

In Discrete dynamics and number theory, one of the most important challenging conjectures is Collatz conjecture. Some basic algebraic and analytical properties of Collatz like integral value transformations (IVTs) in the context of discrete dynamical system over \(\mathbb {N}_0\) are adumbrated. The dynamical maps associated to the dynamical systems are everywhere continuous but nowhere differentiable (smooth) in domain \(\mathbb {N}_0\). Under such Collatz like IVTs, for any initial point \(X_0\) \(\in \) \(\mathbb {N}_0\) the dynamical systems are convergent and converge eventually to a single point attractor, zero. In other words, the image sets associated to the Collatz like dynamical maps in each iteration converge to a single point image set consisting the point zero. The rate of convergence of the Collatz like dynamical system with respect to some property is also described. It is observed that the \(l_2\)-norm of the ith order derivative operator for all Collatz like map is a decreasing convergent sequence which converges to zero. Keywords Discrete dynamical systems Fractal dimension Integral value transformations & Collatz like IVTs