Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 126.08803
Autor: Erdös, Pál; Jabotinsky, E.
Title: On analytic iteration (In English)
Source: J. Anal. Math. 8, 361-376 (1961).
Review: If F(z) = z+summ+1oo fkzk,fm+1 \ne 0, m \geq 1 is any formal power series, then for every complex s there is a unique power series F(s,z) = z+sfm+1(s)zm+1+summ+2 fk (s)zk satisfying F(F(s,z)) = F(s,F(z)). Moreover F(1,z) = F(z), F(s,F(t,z)) = F(s+t,z), and fk(s) is a polynomial in s. If F(z) and Fs(z) have a positive radius of convergence, F(s,z) is called an analytic s-iterate of F(z). The authors investigate the nature of the set S of s-values for which F(s,z) is an analytic s-iterate of F(z). It is clear that if S contains a real neighbourhood of s = 0, then it contains the whole real axis R. The authors show that if R belongs to S, then S is the whole complex plane. Further, if S is not the whole plane, then S has plane measure zero and S \cap R has linear measure zero. This work was extended by the reviewer (Zbl 113.28302) and by the second author (Zbl 113.28303).
Reviewer: I.N.Baker
Index Words: complex functions
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