Periodic Point at Real Axis for a Generalized 3x+1 Function

Abstract:

In order to study the fractal character of representative complex exponential function just as generalized 3x+1 function T(x). In this essay, we proved that T(x) has periodic points of every period in bound (n, n+1) when n>1 in real axis. Then, we found the distribution of 2-periods points of T(x) in real axis. We put forward the bottom bound of 2-periodic point’s number and proved it. Moreover, we found the number of T(x)’s 2-periodic points in different bounds to validate our conclusion. Then, we extended the conclusion to i-periods points and find similar conclusion. Finally, we proved there exist endless convergence and divergence points of T(x) in real axis.

• Periodical: Applied Mechanics and Materials (Volumes 55-57)
• Pages: 1670-1674
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.55-57.1670
• Online since: May 2011
• Authors: Shuai Liu, Zheng Xuan Wang
• Keywords: Fixed Point, Fractal, Generalized 3x+1 Function, Periodic Point, Real Axis

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