G. Szekeres, Regular iteration of real and complex functions, Acta Math., 100 (1958), pp. 103--258. 185  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of a given size, so that we need to hold the size fixed and vary the height. The first results of this kind which involved nonlinear analytic iteration were obtained by Renyi and Szekeres [34] in a study of rooted nonplanar labeled trees. By means of an extensive study (which relied heavily on [36]) of the sequence of functions G 0 (z) G 1 (z) where G 0 (z) z and G h 1 (z) z exp(G h (z) h 0 , 3.1) they showed that the average height of rooted nonplanar labeled trees with a nodes is asymptotic to (2pn) 1 2 as n . Furthermore, Renyi and Szekeres obtained the ....

....z for which Q h (z) as h , and it was necessary to obtain delicate estimates for the rate of divergence. The proof of Theorem 4.1, on the other hand, deals only with those z for which Q h (z) 0 as h . The literature on the general subject of iteration of rational maps is immense (cf. [1,5,9,12,20,23,36]) and constantly growing, especially because of the new impetus given to that subject by work in mathematical physics. Some of that work is closely related to the investigations that were undertaken in connection with tree enumeration. From the classical results of Fatou [12] and Julia [20] it can ....

G. Szekeres, Regular iteration of real and complex functions, Acta Math., 100 (1958), 203-258.

.... is, Abel s functional equation for h (u) Solution of this equation is by no means unique, not even if we disregard the obviously free additive constant; however there is a unique analytic solution (up to an additive constant) with an appropriate regular asymptotic behaviour when u 0 (see [26]) Our problem can now be rephrased as follows: find 0 ff b 1 and an analytic f : 0; b) 0; 1) such that h (u) is defined through (12) and (13) and A(u) through (14) then A(u) is the regular solution of (A) We cannot drop the regularity requirement; it would conflict with the ....

....would conflict with the monotonicity of the solution near b. The form of the regular solution of (A) depends on the Taylor coefficients of h (u) and in particular on the fact that the coefficient of u 2 in (13) is 0. Assuming that b 3 6= Gammab 2 2 , that is, a 3 6= 0, A(u) has the form ([26], p.254) A(u) c Gamma2 =u 2 c Gamma1 =u c 0 log u c c 1 u c 2 u 2 Delta Delta Delta : 15) The coefficients c k can be obtained by formal substitution of (13) and (14) into (A) In particular c Gamma2 = Gamma 1 2a 3 ; c Gamma1 = a 4 a 2 3 ; c 0 = 3 2 Gamma a 5 ....

G. Szekeres, "Regular iteration of real and complex functions," Acta. Math. 100, 203-258, 1958.

....The series (2 1) converges, by a century old theorem of Konigs [1884] see [Kuczma 1968, Chapters VI and VII] for results of this kind. The series (2 2) is not necessarily convergent; nevertheless there is a unique analytic solution of (1 1) having (2 2) as an asymptotic expansion at x = 0 [Szekeres 1958]. Although the principal Abel function is the one that behaves best near 0, there is no reason to assume that it is also the solution that has the most regular manner of growth at x 1. Let D(x) 1=A 0 (x) denote the reciprocal of the derivative of the principal Abel function. If c 1 it ....

G. Szekeres, "Regular iteration of real and complex functions", Acta Math. 100 (1958), 203--258.

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G. Szekeres, Regular iteration of real and complex functions, Acta Math., 100 (1958), pp. 103--258. 185

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