The Existence of Fixed Point for a Generalized 3x+1 Function
In order to study generalized 3x+1 function C(z), we find the character of fixed points of C(z) at real axis by complex analytical analysis. Then we improve the solving algorithm of its fixed points. We proved that the integer fixed points of C(z) are 0 and -1 and the attract fixed points of C(z) are 0 and -1.2777. Popularized the result to complex plane, we prove that there is no fixed point of C(z) except real axis. We draw the fractal figures of C(z) by escape time algorithm to prove the result and give a conjecture from the fractal figures. The conjecture guess that the attract domain of C(z) is a connected domain which is connected with single point.
S. Liu et al., "The Existence of Fixed Point for a Generalized 3x+1 Function", Applied Mechanics and Materials, Vols. 55-57, pp. 1341-1345, 2011