Half-iterates and periodic stuff , my mod method [2019]
consider complex numbers z and an analytic function f(z) such that all values f(z),f(f(z)),... belong to the same siegel disk and all iterates f^[n](z_0) ( for any fixed z_0) are distinct. 
the iterations of f on starting values z are thus iso with rotations on a unit circle with irrational period.

to compute the half-iterate of f for z_0 we use the following method below based on modular arithmetic and many iterations.

first we must compute the period.  

we pick the most logical period.

let me explain

consider the unit circle. 

assume the period is p > 1. 

- p is then also a period. 

if we assume a single iteration always goes around the circle , then our period becomes p/(p+1) instead of p.
let me explain that :
1 mod p is the normal iteration "1" time and p is the period p > 1.

if we reconsider and let 1 iteration go around the circle once then we get instead :

1 + p mod p

divide both sides by 1+p

1 mod p/(p+1)

so our new period ( p') is then p/(p+1).

p' = p/(p+1) 

those periods are relatively the same. they give the same integer iterates , however they differ for computing half iterates ... because

1 + p mod p can be considered as 1 + p = 1 + 0 = 1 mod p and then the natural iterations agree. 

but 1\2 mod p is NOT the same as (1+p)\2 mod p in the sence that p/2 is a half rotation of the circle.

so (1+p) /2 =  1/2 + p/2 mod p ( a half iterate + a half rotation !! )

this is clearly different.

notice p/(p+1) < 1 ! and also the half-iterate f^[1/2](z_0) would then not be between z_0 and f(z_0).

so we only consider irrational periods p > 1.

( you might want to think about that before you continue )

next assuming the period is not given and/or we need good rational approximations, we use the following :

lets take T iterations , T stands for trials.

count the number S , S stands for succes , where S is the amount of " succes ".
succes is defined as getting values f^[j](z_0) ( from the T iterations ) between z_0 and f(z_0).

for instance assume we take 6 iterations (f(z_0),f(f(z_0)),...) and we have 1 succes then the period approximation is T/S = 6.

now take as many T until S is of the form S = 2^m.

so we get the rational approximations T_m/S_m = T_m/2^m.

we need an extra constraint here , T and S are relative prime ( reduced fraction ! ) so we need

T_m = 2s + 1.

notice T/S > 1 btw. ( consistant with p > 1 too ) because T > S.

taking all that into account we get T_k / S_k = (2s + 1)/2^k.

so recap

a single iteration is " 1 " ( or 1 mod p ) and the good ( binary ) period approximations are T_k/S_k.

v iterations are "v" ( or v mod p ).

we need to find integer v approximating the half integer iterate.

this is where mod comes in.

1 mod T/S 
 iso to one iteration.

we want to find 

v = 1/2 mod T/S.

start with 

1 mod T/S

multiply both sides by S

S mod T

now we want

S v mod T = S/2 mod T

we know S is even so the half iterate is between z_0 and f(z_0).

v = 1/2 mod T

T is odd so

v = (T + 1)/2. 

so our succesive approximations are the positive integers v_k :

v_k = (T_k + 1)/2.

and we use them to get closer and closer to the half iterate.


let the period be p.

notice f^[p](z) = z = id(z).

by analytic continuation we get for any complex number G ,  f^[p](G) is id(G) !!! also for other complex values d not on the siegel disk !!

so no analytic solution of ( or on ) the siegel disk can be considered an analytic solution of fractional iterates of f in general.

We also know that agreement on fixpoint and cyclic points is rare.

similar problems occur with herman rings.

and perhaps attracting periodic orbits ( attracting siegel or herman ) lead to similar problems.

It seems therefore logical to assume connected julia sets of f [, which contains these siegel disks , herman rings , attracting periodic orbits , fixpoints and cyclic points in a dense way ( together with the even more problematic self-intersecting type chaotic orbits ) ] are to be avoided for fractional analytic iterations.
in fact they seem logical boundaries for analytic continu iterations , in the sense that they cannot both be analytic AND satisfy the equations.

What remains not so clear to me is the case of nowhere connected julia sets.

i use connected and continu as synonyms here. 

I would like to add one more thing :

as far as i know , it is still unknown in general whether the julia set of newtons method for entire transcendental functions are connected.

i think it was unsolved until at least 2009 and I am unaware of a solution today (10 years later ).


this might also explain why methods based on cyclic or periodic concepts usually are not succesful in complex dynamics. 

i assume this is not only the case for analytic solutions but also C^oo ones.


I am aware Gottfried posted about fractional iterations for closed orbits ( siegel disks ) recently , but I think we see or do things differently. or maybe not. 
I posted this to show my methods and ideas about them.

maybe i just do not understand his posts. 
in particular his sum related to continued fractions of the golden mean. 


although my idea and methods seem solid there might be things to argue and consider. 

for starters , is this the complete story for periodic nonintersecting orbits ?

do we really have converge to the correct value ??

uniform limits ?

what about " misbehavior " periods , such as pisot numbers or brjuno numbers ? 
or just complicated continued fractions ?

How does all this relate to all previous stuff about tetration and Dynamics ?



A mod B is equivalent or iso to AC mod BC might be controversial , but it works.

i call it tommy's mod rule. 

maybe this has already been named ? 

inform me.


i havent read or posted in a long time. I was absent doing other things. other math too. and chess.

sorry about that.

did i miss alot ? 


thanks to ( mainly ) gottfried and sheldon to get me back here again.

i hope i do not duplicate old ideas of myself or others. 

my apologies for not using caps or tex , I am - as usual - in a hurry and typed this in a few minutes.

this post is not reviewed or edited and might be informal. 
but I think the general idea is clear and worth posting here. 


I might post more since i have infinite ideas but finite time. 




" truth is what does not go away when you stop believing in it "

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