[UFO] Attracting Fixpoints or attracting line?

[tommy1729]

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 \( k\in\mathbb{Z} \).

well , Bo , apart from the " no " i agree on what you said.

in fact i already understood that.

however , isnt t_k the same for e.g. k = 2 and k = 4 in the fixed points of exp( t_k + k2pi i ) = t_k ?