Coloring F(x) = F(exp(x)) tommy1729 Ultimate Fellow     Posts: 1,370 Threads: 335 Joined: Feb 2009 05/10/2015, 08:41 AM (This post was last modified: 05/10/2015, 04:05 PM by tommy1729.) Consider a 1-periodic real-analytic function of the real-analytic slog for re > 0. That satisfies : F(x) = F(exp(x)) However this functional equation can not hold for all nonreal x because exp is a chaotic map. So we color the complex plane according to how many times the equation holds, more precisely : Black x F(x) =\= F(exp(x)) And F(x) =\= F(exp^(x)) Purple x F(x) =\= F(exp(x)) And F(x) = F(exp^(x)) Blue x F(x) = F(exp(x)) And F(x) =/= F(exp^(x)) Red x F(x) = F(exp(x)) = F(exp^(x)) Tommy's 4 color conjecture F is completely determined by its coloring and F(1). Not to be confused with the 4 color theorem Regards Tommy1729 tommy1729 Ultimate Fellow     Posts: 1,370 Threads: 335 Joined: Feb 2009 05/10/2015, 04:16 PM I corrected the title and first post. Im a bit confused by this : If F is piecewise analytic and F(x) = F(exp(x)) Then F(x) - F(exp(x)) = g(x) And g(x) should be piecewise analytic too. But g(x) = 0 locally. However 0 has no analytic continuation apart from being 0 EVERYWHERE. This seems like a contradiction. I might have asked this before srr. Regards Tommy1729 tommy1729 Ultimate Fellow     Posts: 1,370 Threads: 335 Joined: Feb 2009 05/14/2015, 09:49 PM Post 2 is simple. G has a singularity when slog has. The functional equations holds when g = 0. The question is Let ln(ln(s)) = s such that ln(s) =\= s. What is slog(s) ? And slog(exp(s)) - slog(s) ? Maybe naive but if n > 1 and n is the smallest n such that Ln^[n](d_n) = d_n Then the naive conjecture is Slog(exp^[m](d_n)) - slog(d_n) = m mod n. Weak version : m positive integer. Strong version : m positive real. Variants : negative or complex. Regards Tommy1729 tommy1729 Ultimate Fellow     Posts: 1,370 Threads: 335 Joined: Feb 2009 05/14/2015, 10:08 PM The naive conjecture seems to ignore 2 pi i. Thinking ... Regards Tommy1729 nuninho1980 Fellow   Posts: 95 Threads: 6 Joined: Apr 2009 05/14/2015, 10:35 PM STOP due to the addiction!!!  tommy1729 Ultimate Fellow     Posts: 1,370 Threads: 335 Joined: Feb 2009 05/14/2015, 11:05 PM (05/14/2015, 10:35 PM)nuninho1980 Wrote: STOP due to the addiction!!! ??? nuninho1980 Fellow   Posts: 95 Threads: 6 Joined: Apr 2009 05/14/2015, 11:19 PM (05/14/2015, 11:05 PM)tommy1729 Wrote: ???You should be (much) slower to reply. « Next Oldest | Next Newest »