Dynamics analysis for a generalized approximate 3x + 1 function

Abstract

To study character of the generalized 3x + 1 function is a difficult problem in fractal. At first, we put forward a generalized approximate 3x + 1 function D(z) and point out fractal character of D(z) is similar to generalized 3x + 1 function. Secondly we execute dynamics analysis of D(z) and find the fixed point and periodic point. Thirdly, we point out all fixed points are at real axis of complex plane. We proved there is no other attract fixed point except zero and find the distribution of periodic point in complex plane. Then we proved that iteration of D(z) is not divergence at whole number. Finally, we draw fractal figures to validate the dynamics character of D(z). © 2012 Springer Science+Business Media B.V.