An Asymptotic Formula for the Iterates of a Function

Abstract

Let IK be either IR or ℂ and D an open set of IK containing 0 and starlike with respect to 0 (i.e. an open interval containig 0 in the case IK = IR). If f: D » IK is a continuous function with fixed point 0, then under certain conditions stated below we can prove for the kn- th iterates of f the following asymptotic formula:

$$f^{(kn)}\bigg({x \over n}\bigg )=\sum_{i-1}^r{1\over (nk)^i}\ f_i(kx)+o \bigg({1\over n^r}\bigg),$$

(1)

for n » ∞, k, n and r beeing positive integers and x close enough to 0. The functions f _i are continuous and uniquely determined by f.

In particular (1) holds for any function holomorphic on a neighbourhood of zero, having a convergent power series expansion of the form

$$f(z)=z+a_2z^2+\cdots=\sum_{j=1}^\infty\ a_jz^j,\ a_j\in {\cal C},a_1=1,$$

and for any integers k, r with r > 0.