04/09/2010, 06:43 AM
(04/08/2010, 10:03 PM)tommy1729 Wrote: your plot gives the fixed points of exp( t_k + k2pi i ) = t_k
but should that not be periodic with k ?
since exp( x + k i ) is periodic with real k ?
im a bit confused now ...
No, the fixed points of exp, i.e. the points with exp(z)=z, can be obtained by considering \( z=\log(z)+2\pi i k \), \( k\in\mathbb{Z} \) i.e. the fixed points of the branches of the logarithm. All the non-real fixed points of exp are repelling, i.e. |exp(z)|>1, thats why these fixed points are attracting for the logarithm and can be obtained by iterating the corresponding branch.
Gottfried's question was now what happens if we choose non-integer \( k \); and found that the corresponding "fixed points" lie smoothly on a line between the proper fixed points of \( k\in\mathbb{Z} \).
