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+--- Thread: [2014] exp^[s](z) = z (/showthread.php?tid=916)
[2014] exp^[s](z) = z - tommy1729 - 08/26/2014
Consider the equation for a given real s > 0 :
exp^[s](z) = z
such that |z| < exp^[s](s).
Let the number of distinct solutions be T[s].
Because of the fact that if w satisfies
exp^[s](w) = w
then conj(w) = w_ satisfies
exp^[s](w_) =w_
Hence we can conclude that T[s] is Always even.
Therefore I am intrested in T[s]/2.
I would love to see plots and tables of s vs T[s]/2.
Many conjectures are possible.
Probably connected to fractal theory , basic dynamical systems and bifurcations.
Being very optimistic I would say T[s]/2 might have a closed form.
Possible conjectures could look like :
1) T[s]/2 = O ( s^a b^s ) for some real a,b.
2) Let p be an odd prime such that p+2 is not a prime.
Then T[p] =< T[p+2].
Perhaps someone here is an expert on these things ?
Let n be a positive integer.
Could T[n] satisfy a recursion ?
Like T[2n] = T[2n-1] + T[n] :p
regards
tommy1729