The link on extraneous non-repelling cycles of Schröder's methods of the first and second kind

Abstract

Let Sf,n and Kf,n be the functions defined in Schröder's method of the first and second kind for an entire function f with given order n(n≥2), respectively. Based on unrefined algebra characterizations of Sf,n and Kf,n, we obtain some sufficient conditions on f such that both Sf,n and Kf,n possess given finite pairs of extraneous non-repelling cycles. Here, these conditions are a pair of equations, which have infinitely many polynomials or transcendental entire functions as its solutions. For obtaining some solutions f of such equations, we provide a step-by-step method. We start from any solution g in corresponding equations so that the function Sg,2 possesses the above finite pairs of extraneous cycles but all are super-attracting, and then f can be obtained by a series of formulas concerning the function g, points and multipliers of those cycles. Note that Sf,2=Kf,2 is identical with Newton's method for f. In a sense, this fact reveals that some extraneous super-attracting cycles of Newton's method imply certain extraneous non-repelling cycles of any method from the two families of methods. More generally, for any given orders n and m, some extraneous non-repelling cycles of Sf,n or Kf,n imply that of SF,m or KF,m for some entire functions f and F. These give a partial answer for the problem of finding possible link between the two families of methods, which was posed by Steven Smale in 1994.