Fixed Point and Fractal Images for a Generalized Approximate 3x+1 Function

In order to study the fractal character of complex exponential functions represented by generalized 3x+1 function T(x),a generalized approximate 3x+1 function B(x) is constructed.With complex analytical analysis for B(x),the fixed points of B(x) at real axis and C-plane are found.It is proved that zero is the only attract fixed point and the iteration of B(x) on the complex plane is symmetrical.And the domains of the constringency and the divergence of B(x) are given.Finally the fractal images are drawn to validate the fractal character of B(x) and a conjecture,which the constringency and the divergence of B(x) is nested,is put forward.