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 twice a superfunction MphLee Fellow Posts: 95 Threads: 7 Joined: May 2013 03/24/2014, 09:12 PM (08/11/2010, 04:28 PM)tommy1729 Wrote: i have been thinking about the following often : how and when is a function a 'superfunction for two functions' and what degrees of freedom do we have ? for instance assume the following equation f(x+1) = A(f(x)) f(x+i) = B(f(x)) and A and B are somewhat related : they commute of course and satisfy some equation ( e.g. A = sin(log(B)) ) i assume there is no freedom for f(z) apart from choosing f(0). ( once again , riemann mapping theorem and double periodic functions convinced me of that ) but its not so clear when we have a solution and when not. also , could this be a usefull uniqueness criterion ? its seems we need at least 3 criterions for a solution to exist : 1) A and B commute 2) A and B share the same fixpoints 3) its clear the superfunction of A and B is nonparadoxal at complex oo. maybe 4 : 4) the superfunction becomes periodic or semi-periodic near complex oo. ( although that might follow from 3 ) regards tommy1729 $A(x)=f(f^{\circ -1}(x)+1)$ and $B(x)=f(f^{\circ -1}(x)+i)$ but this doesn't mean that $A^{\circ i}(x)=B(x)$ should hold always? MathStackExchange account:MphLee « Next Oldest | Next Newest »

 Messages In This Thread twice a superfunction - by tommy1729 - 08/11/2010, 04:28 PM RE: twice a superfunction - by MphLee - 03/24/2014, 09:12 PM RE: twice a superfunction - by tommy1729 - 03/25/2014, 12:34 AM RE: twice a superfunction - by MphLee - 03/25/2014, 08:51 AM RE: twice a superfunction - by tommy1729 - 03/26/2014, 01:23 PM RE: twice a superfunction - by MphLee - 03/26/2014, 03:34 PM

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