conjecture 666 : exp^[x](0+si)
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 :

\( bound(2)=666.141.. \)


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 ?


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,

N(\mathcal{N},L,\epsilon)= n\\

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 



Attached Files
.pdf   mandel.pdf (Size: 205.11 KB / Downloads: 206)

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