From: "G. A. Edgar" Subject: Re: f(f(x)) = g(x) Date: Mon, 28 Aug 2000 08:58:31 -0400 Newsgroups: sci.math Summary: [missing] From an old post by Zdislav V. Kovarik: Marek Kuczma: Functional Equations in a Single Variable Monografie Matematyczne 46, Warsaw 1968 In Ch. XV, Sec. 6, he writes [modified for ASCII format]: "For the equation (*) f^2(x) = e^x, a real analytic solution has been found by H. Kneser. This solution, however, is not single-valued (Baker) and, as pointed out by G. Szekeres, there is no uniqueness attached to the solution. It seems reasonable to admit f(x)=F^(1/2)(x), where F^u is the regular iteration group of g(x)=e^x, as the "best" solution of the equation (*) (best behaved at infinity). However, we do not know whether this solution is analytic for x>0. [Kuczma defines and discusses regular iterations at infinity in Chapter IX, Sec 5.] References: Baker, I.N.: The iteration of entire transcendental functions and the solution of the functional equation f(f(z))=F(z), Math. Ann. 120(1955), pp. 174-180 Kneser, H.: Reele analytische Loesungen der Gleichung f(f(x))=e^x und verwandten Funktionalgleichungen, J. reine angew. Math. 187(1950), pp. 56-67 Szekeres, G.: Fractional iterations of exponentially growing functions, J. Australian Math. Soc. 2(1961/62), pp. 301-320 -- Gerald A. Edgar edgar@math.ohio-state.edu ============================================================================== From: Pierre Bornsztein Subject: Re: f(f(x)) = g(x) Date: Mon, 28 Aug 2000 15:55:33 +0200 Newsgroups: sci.math Jonathan Hoyle wrote: > > I posted this question some months ago with limited response, so I > figured I'd try it again. > > Is there any investigation on decomposing function iterations? > Specifically, if f(x) is any continuous function, and we define > inductively: > > f1(x) = f(x) > fn(x) = f(fn-1(x)) > > Given the equation fk(x) = g(x) for some fixed k and function g, can one > determine if there is a solution for f? Particularly looking at k = 2. > > For f(f(x)) = x, several solutions come to mind: x, -x, 1/x. For g(x) a > linear equation ax+b, we have the solutions f(x) = sqrt(a)*x + > (b/(1+sqrt(a))) and its negative. > > But what about g(x) a quadratic? Let g(x) )ax^2 + bx + c It has been proved [1] that : 1) For a,b,c are real numbers and g:|R --> |R - if (b - 1)^2 - 4ac > 1, then for every integer r > 1 there does not exist a function f:|R --> |R such that f^r = g, where f^r denotes the r°iteration of f. - if (b - 1)^2 - 4ac =< 1, then for every integer r > 1 there exists a function f:|R --> |R such that f^r = g. 2) For a,b,c complex numbers and g:C -->C, then for every integer r > 1 there does not exist a function f:C-->C such that f^r = g. [1] R.E.Rice,B.Schweizer, A.Sklar "When does f(f(z))= az^2 + bz + c?" Amer.Math.Monthly (1980) p.252-263. Pierre. ============================================================================== From: Andrew Osbaldestin Subject: Re: f(f(x)) = g(x) Date: Sat, 02 Sep 2000 15:32:02 +0100 Newsgroups: sci.math,sci.fractals The problem of "functional square roots" is an old and fascinating one. For the specific case of finding f such that f(f(x))=exp(x) there are fact uncountably many solutions. One method I know is to pick an arbitrary negative real number -c and an arbitrary increasing function g on (-infinity,-c] going from -c to 0 and "bootstrap" to define g on the rest of the real line. For instance, the next interval it gets defined on is [-c,0], its values enforced by requiring g(g(x))=exp(x) on the first interval, (-infinity,-c]. I think H. Kneser in the 1930s(?) proved that there is no complex analytic functional square root of the exponential. The fact that exp(x) is monotonic makes life easy. I don't know off hand about nonmonotonic functions (such as x^2 or sin(x)). There are a couple of nice textbooks on "iterative functional equations" by M. Kuczma. The Cvitanovic'-Feigenbaum functional equation was first shown to have a solution by O. Lanford in 1982 who used a computer assisted proof. Subsequently proofs "by hand" have been given by H. Epstein and C. McMullen. These functional equations which relate composition and scaling appear to be a much deeper kinds of problems than that of functional square roots. Andy Osbaldestin -- in article 39AC0E3F.8CFE6809@earthlink.net, ROGER BAGULA at tftn@earthlink.net wrote on 29/8/00 7:26 pm: > Cvitanovic'-Fiegenbaum equation on page 71of Ruelle's > "Elements of differentiable Dynamics and Bifurcation theory": > g(g(a*x))+a*g(x)=0 > let y=a*x > g(g(y))=-a*g(y/a)=f(y) : ( your equation with f for g) > These are involved with renormalization on the disk |x|<1 > and g has an hyperbolic fixed point. The derivatives are > contained in the unit disk. > The give example of equations like the Julia: > f(x,m)=1-m*x^2 > with m=1.401155. > McMullen has done a paper on this kind of renormalization, I think. > > Jonathan Hoyle wrote: > >> I posted this question some months ago with limited response, so I >> figured I'd try it again. >> >> Is there any investigation on decomposing function iterations? >> Specifically, if f(x) is any continuous function, and we define >> inductively: >> >> f1(x) = f(x) >> fn(x) = f(fn-1(x)) >> >> Given the equation fk(x) = g(x) for some fixed k and function g, can one >> determine if there is a solution for f? Particularly looking at k = 2. >> >> For f(f(x)) = x, several solutions come to mind: x, -x, 1/x. For g(x) a >> linear equation ax+b, we have the solutions f(x) = sqrt(a)*x + >> (b/(1+sqrt(a))) and its negative. >> >> But what about g(x) a quadratic? Or g(x) = sin(x)? Or e^x? >> >> Thanks in advance. >> >> Jonathan Hoyle >> Eastman Kodak > > -- > Respectfully, > Roger L. Bagula > tftn@earthlink.net > 11759Waterhill Road > Lakeside,Ca 92040-2905 > tel: 619-5610814 > URL: http://home.earthlink.net/~tftn > URL: http://www.geocities.com/ResearchTriangle/Thinktank/7279/ > URL: http://www.angelfire.com/ca2/tftn/index.html > URL: http://members.xoom.com/RogerLBagula/index.html > URL: http://sites.netscape.net/rlbtftn/index.html > URL: http://victorian.fortunecity.com/carmelita/435/ > URL: http://members.tripod.com/tftnrlb/index.html > Chat Room URL:http://planetall.homestead.com/tftn/index.html > URL: http://fractals.jumpfun.com/ > URL: http://members.spree.com/education/tftn9/ > URL: http://www.crosswinds.net/~translight/index.html > URL: http://www.freestation.com/ca/tftnroger/index.shtml > >