Jeffrey P. Dumont and Clifford A. Reiter, Visualizing Generalized 3x+1 Function Dynamics, Computers & Graphics, 25 5 (2001) 883-898.

The function that results in 3x+1 for odd integers x and half of x for even x has led to intriguing questions. The 3x+1 conjecture states that iteration of that function on positive integers eventually results in the value 1. We investigate many generalizations of that function to the complex domain and visualize the resulting dynamics using escape time, stopping time, and basin of attraction images. We will see beautiful, rich dynamics consistent with the conjecture. Two of the most interesting generalizations that we consider are of the form T(x)=(1/2)(3mod2(x)x+mod2(x)) where mod2(x) is either sin2(pi x/2) or 1/2(1-epi i x). We see that some generalizations have relatively simple real dynamics, which may make them useful for analysis and we see a complex generalization where sequences of stable egg shaped regions appear in coefficient stopping time images that suggests remarkable patterns for such stopping time.

See also: · Some Images and animations related to the paper [Images and animations] · Paper [preprint]. · A paper exploring the dynamics of one generalzation of the 3x+1 function [dy_2003_a] · Jeff Lagarias' 3x+1 page with papers and annotated bibliography [Lagarias 3x+1] .