Tetration Forum

Many many years ago I conjectured the following :

Consider exp iterations of a starting value $y=0+si$ for $0<s<2$ .

It seems for all such s , after some iterations we get close to 0.

So we search for positive real x > 1 , such that

$exp^{[x]}(y)=0$

or it gets very close.

I believe an upper bound for x is

$bound(x)=1+exp(s^2+s+1/s)$

I gave it the funny name conjecture 666 because by the formula above :

$bound(2)=666.141..$

$Smile$

Notice the formula does not extend correctly to values such as $s=\pi$ since that sequence will never come close to zero.

A sharper bound is probably attainable.
But how ?
Is there an easy way ?

For 1/2<s<2 this might be achievable by computer search ? And/or calculus ?

But is there an easy or short proof ?

Notice that taking derivatives of exp(exp(... is not easier than computing exp(exp(... so it seems hard to shortcut the problem.

I was inspired to share this idea of mine here because of memories of some people on sci.math.

What do you guys think ?

Regards

tommy1729

I believe Milnor has something to say about this, I'll try and check his book later.

The gist is, that $\exp^{\circ n}(z)$ gets arbitrarily close to $0$ for almost all $z$ (it misses on a Lebesgue measure zero set (which are the periodic points)). I believe Milnor has a section discussing how to estimate the size of $n$ ; particularly, if I recall correctly; it'll give a nice bound on the size of $n$ in relation to its nearness to 0... It's something to do with Julia sets; I can't remember perfectly.

I can't remember if it's Milnor though; or I read it somewhere else. I'll have to look through the book again...

EDIT: ACK! I'm having trouble finding it; but from memory, there's a word for it. Consider $f:\mathbb{C}\to\mathbb{C}$ . Take a neighborhood $\mathcal{N}$ about a point $z \in \mathcal{J}(f)$ (the julia set of $f$ ). Then the orbits of $f$ on $\mathcal{N}$ are dense in $\mathcal{J}(f)$ . From here you can create a function,

$<br /> N(\mathcal{N},L,\epsilon)= n\\<br />$

Such that some point of $z' \in \mathcal{N}$ satisfies $|f^{\circ n}(z') - L| < \epsilon$ for some $L \in \mathcal{J}(f)$ . (or something like this.)

I believe there is a way to estimate this function; I can't seem to find it at the moment. I'll keep looking.

It would be very useful to have such an estimate formula !!

The problems reminds me of " the opposite question " where we measure when we get " away " , rather then within boundaries as my conjecture studies ;

see attachement

regards

tommy1729

tommy1729

JmsNxn

tommy1729