Date: Tue, 7 Apr 1998 19:08:48 -0400 (EDT) From: "Zdislav V. Kovarik" To: Dave Rusin Subject: Re: Smooth log^* function? The Abel equation can be solved for a modification of the logarithm function, namely for f(x) = ln(1+x) ; luckily, f has 0 as a semi-stable fixed point. The solution A(x) satisfies A(ln(1+x)) = 1 + A(x). (Kuczma uses lower case German "a" instead of A. Or is it the inverse that he denotes so?) This A can be obtained by a limiting process, and it helps extend the iteration group (from integers) to a composition group from the reals. The material is scattered all over the book Functional Equations in a Single Variable. by Marek Kuczma, Polish Scientific Publishers 1968 and there is a newer book Iterative Functional Equations, by Marek Kuczma Bogdan Choczewski and Roman Ger Cambridge University Press 1990 ISBN 0521 355613 In Kuczma's book, there is a bibliographical reference to a true iterated exponential function, with caveats about its analyticity or lack of it. Sorry I kept postponing the answer, Slavek Kovarik. On Fri, 5 Dec 1997, Dave Rusin wrote: > In article <667jag$rm3@mcmail.CIS.McMaster.CA> you write: > >In article <3485D35E.1320E2AA@cs.utexas.edu>, > > >How to find a k-times iterated function f, where k is a real parameter: If > >possible, find a one-to-one smooth solution A of Abel's equation > > > > A(f(x)) = 1 + A(x) > > > >and then > > > > f(iterated k times)(x) = A^(-1) (k + A(x)) > > The poster wanted to do this with f = exp. Are you suggesting it would be > possible to describe a smooth A in this case? I've given this some > thought before and to my frustration couldn't find a reasonable description > of such an A (that is, no natural differential equation it should > satisfy or anything). Have I overlooked something simple? > > dave