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

[tommy1729]

Many many years ago I conjectured the following : 

Consider exp iterations of a starting value for .

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

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



or it gets very close.

I believe an upper bound for x is 



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





Notice the formula does not extend correctly to values such as  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

[JmsNxn]

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

The gist is, that gets arbitrarily close to for almost all (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 ; particularly, if I recall correctly; it'll give a nice bound on the size of 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 . Take a neighborhood about a point (the julia set of ). Then the orbits of on are dense in . From here you can create a function,



Such that some point of satisfies for some . (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.

[tommy1729]

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