Attraction in positive direction of real axis of two generalized 3x + 1 functions

Abstract

To study attractive character of two generalized 3x + 1 functions T(z) and C(z) at positive direction of real axis, at first, we define attractive domain and series. Secondly, we analyze attractive domain and series of T(z) and C(z) and point out the structure of attractive domains surrounding the same attractive series that are similar to each other. Finally, we draw the fractal figures of the two functions by escape time algorithm and validate the structure rule of the attractive domain and series of T(z) and C(z) from the figures. © 2012 Springer Science+Business Media B.V.