Zeta iterations

[tommy1729]

just because the absolute value of the derivative at z is smaller than 1 is that reason

I see, but how did you derived that? Is it very straightforward from the definition of attractive fixed point but I cannot see it? I am having a feeling that I possibly asked a pretty stupid question

I wanted to comment on that to give some kind of intuitive explaination.

For simplicity recenter the fixpoint at 0.

Now if f(z) is analytic and f(0)=0 and there is no fixpoint near 0 then consider writing f(z) = 0 + a z + b z^2 + c z^3 + ...

Now from calculus you know that if z gets very small ( where very depends on the size of the taylor coefficients and the remainder ) then f(z) gets well approximated by f(z) = a z + O(z^2)

Now if a is 1 then f(z) approximates id(z) and hence the fixpoint 0 is " neutral " also called " invariant " or " parabolic ".

Now clearly if a = i ( imaginary unit) then by f(z) = a z + O(z^2) we get a simple rotation but not a strong attraction or repellation.

This already hints that looking at the complex number a in polar coordinates makes more intuitive sense.

To visualize the 'small' z take a disk around z=0 with a small radius.

Now because f(z) approximates a z , AND the theta of a determines how much we rotate , the absolute value must determine if it is attracting or repelling.

By attracting we mean that any point in the small disk must be mapped closer to the fixpoint , and by repelling we mean that any point must be mapped further away from the fixpoint.

For instance if f(z) is of the form 0 + 0.5 z + 0.0001 z^2 + ... then it is clear that f(z) in the small radius ( = z small => higher powers of z have lesser influence ) behaves like f(z)^[t] = 0.5^t z which is clearly attracting.

By analogue if a = 2 it is clearly repelling.

if a = 2 i , then that is equivalent to repelling + rotating.

This more or less concludes my personal nonformal intuitive explaination ( for complex values ) and I hope it is clear and helpfull to you.

Btw , Note that I did not discuss what happens when a = 0. I only gave a sketch of how and why when the absolute value of a satisfies 0 < abs(a) < oo.

When a=0 many formula's and properties fail anyway and for the case a=0 you need to understand more of the Fatou-flower.

I could say alot more but I do not want to confuse you with harder stuff and you can read up on the subject too. Hence I did not want to post more than that what occurs sponteanously to me.

Together with some basic complex analysis and basic real calculus
I think this will get you much further in the subjects and discussions about tetration and their cruxial details.


regards

tommy1729