SOLUTIONS TO NONLINEAR RECURRENCE EQUATIONS

Abstract

Let F(z) be any function. Suppose that w is a fixed point of F(z), that is, F(w)=w. Then the recurrence equation xn+1=F(xn) for n=0,1,2,… has a solution of the form xn(w)=w+∑i=1∞a1iAiF•1(w)in, where F•1(z)=dF(z)∕dz. So, for each w there is a set of complex x0 such that x0(w)=x0. We assume that F(z) is analytic at w. This solution appears to be new, even for such famous examples like the logistic map and the Mandelbrot equation.