2.86295 + 3.22327 i Gottfried Ultimate Fellow     Posts: 852 Threads: 126 Joined: Aug 2007 03/07/2021, 01:54 AM (This post was last modified: 03/07/2021, 02:13 AM by Gottfried.) Perhaps you remember the earlier posted reference to my MSE-discussion on periodic points: https://math.stackexchange.com/questions...function-z A better readable pdf-version is attached. I have discussed occurences of periodic points of arbitrary period-length, initially for base exp(1), but also played a bit with complex bases, detecting some fine "irregularities" (I feel that is a good label due to limited occurences). Work in progress, but has been suspended since last year. More discussion on MSE: https://math.stackexchange.com/a/3713978/1714 with complex base I'm not sure whether I've handled your specific base in one of my MSE-answers... Gottfried periodic points compact.pdf (Size: 309.84 KB / Downloads: 204) Gottfried Helms, Kassel tommy1729 Ultimate Fellow     Posts: 1,668 Threads: 368 Joined: Feb 2009 03/07/2021, 11:00 PM thank you for the links Gottfried. As for the rays, using log(log(...log(z)...))) = z probably gives a different dimension , subset or set of rays then the exp counterpart. I assume using the logs gives the smallest solutions. That would make sense to me. *** As for the MSE probably using something that does not have nasty inverses ( unlike like newton method ) is recommended. steffenson maybe ? *** many things can be said about these methods but im still organizing my thoughts. the absolute value of the derivative of exp^[t](z) or z^^t for instance is a major thing. assume z is a cyclic point , is the absolute value of the derivative of exp^[t](z) or z^^t ever = 1 ?? for exp probably i believe not , but for z^^t ?? Now I assumed t integer , but that leads to questions for noninteger. And intersections of iterates etc etc. *** Btw I know you like number theory. My friend mick at MSE posted some of my and his ideas in number theory. What do you think about this one for instance ?? https://math.stackexchange.com/questions...ain-primes regards tommy1729 tommy1729 Ultimate Fellow     Posts: 1,668 Threads: 368 Joined: Feb 2009 03/07/2021, 11:06 PM (03/07/2021, 11:00 PM)tommy1729 Wrote: thank you for the links Gottfried. As for the rays, using log(log(...log(z)...))) = z probably gives a different dimension , subset or set of rays then the exp counterpart. I assume using the logs gives the smallest solutions. That would make sense to me. *** As for the MSE probably using something that does not have nasty inverses ( unlike like newton method ) is recommended. steffenson maybe ? *** many things can be said about these methods but im still organizing my thoughts. the absolute value of the derivative of exp^[t](z) or z^^t for instance is a major thing. assume z is a cyclic point , is the absolute value of the derivative of exp^[t](z) or z^^t ever = 1 ?? for exp probably i believe not , but for z^^t ?? Now I assumed t integer , but that leads to questions for noninteger. And intersections of iterates etc etc. *** Btw I know you like number theory. My friend mick at MSE posted some of my and his ideas in number theory. What do you think about this one for instance ?? https://math.stackexchange.com/questions...ain-primes regards tommy1729 or maybe you like these : https://math.stackexchange.com/questions...c-pin-righ https://math.stackexchange.com/questions...p-2np-2n-1 regards tommy1729 tommy1729 Ultimate Fellow     Posts: 1,668 Threads: 368 Joined: Feb 2009 10/25/2021, 11:50 PM Let X = 2.86295.. + 322327.. i Y = 5 + 9 i then I think using the gaussian method would give nice functions. t(s) = (1 + erf(s))/2 f_X(s) = X^(t(s) * f_X(s-1)) f_Y(s) = Y^(t(s) * f_Y(s-1)) lim n to +oo tet_X(s + x1) = ln_X^[n] f_X(s+n) tet_Y(s + y1) = ln_Y^[n] f_Y(s+n) It feels like the gaussian method might be perfect/fascinating for these bases. the pseudoperiodicity of the X case for instance. (reminds me of the fixpoint methods where the derivative is nonreal ) And maybe easier to compute/plot compared to kneser and/or other bases. And many conjectural ideas seem natural. regards tommy1729 JmsNxn Ultimate Fellow     Posts: 921 Threads: 111 Joined: Dec 2010 10/25/2021, 11:57 PM Hey Tommy, I think you'd like to know, I'm going to try and create a gaussian protocol in my next code. As it works really well at finding non-periodic asymptotic solutions--and perhaps holomorphic final results. I'll try to graph these cases after writing a protocol! Regards, James « Next Oldest | Next Newest » 