[UFO] Attracting Fixpoints or attracting line?

[bo198214]

in fact i already understood that.

No, you still didnt.

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 ?

If you look at my explanations above, you see that the \( 2\pi i k \) is not inside the exponential. The fixed point equation is exp(t_k)=t_k. If you apply the logarithm to both sides you have to deal with the branches as the exponential is not injective on the complex plane.
And these branches introduce the corresponding \( k \).
And yes the fixed points are different for \( k=2 \) and \( k=4 \).
Somwhere on the forum is an animated picture of the fixed points of \( b^t \).