Periodic Point at Real Axis for a Generalized 3x+1 Function
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.
S. Liu and Z. X. Wang, "Periodic Point at Real Axis for a Generalized 3x+1 Function", Applied Mechanics and Materials, Vols. 55-57, pp. 1670-1674, 2011