Jeffrey P. Dumont and Clifford A. Reiter, Real dynamics of a 3-power extension of the 3x+1 function, Dynamics of Continuous, Discrete and Impulsive Systems, 10 (2003) 875-893.

This paper investigates the real dynamics of a generalization of the 3x+1 function using powers of three. Namely, T(x)=(1/2)(3mod2(x)x+mod2(x)) where mod2(x) is sin2(pi x/2). We will see that any cycle of positive integers is attractive for this generalization and the cycle has an expansion factor given by Terras' coefficient function. We will see the function has a negative Schwarzian derivative for x > 0 and will be able to identify invariant intervals and approximately locate the fixed points and critical points. The special simplicity of dynamics around the cycle (1,2) means there is a natural generalization of total stopping time for this function. We conjecture that the odd critical points of this generalization are well behaved. In particular, they lie in the immediate basin of total stopping time surrounding each odd integer.

See also: · Paper [Preprint] · A paper visually exploring the complex dynamics of generalzations of the 3x+1 function [mv_2001c] · Jeff Lagarias' 3x+1 page with papers and annotated bibliography [Lagarias 3x+1] .