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f(s),h(s) and "hairs".
" Hair theory "

Let f(s) be one of those recent compositional asymtotics of tetration.

consider a = f(s_1), b = f(s_2) such that Re(s_1),Re(s_2) > 1.

Now let hair(a),hair(b) be the continu iterations paths(curves) of iterated exponentials of a,b.

So the hair/path/curves starts at a,b and then follow the direction exp^[r](a) or exp^[r](b) for real r >=0.

If f(s) was exactly tetration those paths would be flat and parallel to eachother.

So these hairs must become flatter and flatter as r grows to +oo and exp^[r](a) , exp^[r](b) = f(s_1®) , f(s_2®) 
and Re(s_1®),Re(s_2®) grow also to +oo.
( because we are approximating tetration better and better )

Some questions arise :


When ( if ever ) do such hairs self-intersect ?

And in particular when a,b are close to the real line ?


When does hair(a) intersect with hair(b) ?

if hair(a) eventually becomes the same path (fuse*) as hair(b) or vice versa , this does not count as intersecting.

Again in particular when a,b are close to the real line ?

3) travelling a certain lenght on a hair implies how much iterations of exp ??

In the limit ( r or s going to +oo) this should probably be length 1 implies 1 iteration of exp.

In particular close to the real line.

4) A hair never splits in 2.

5) two hairs with starting points f(s_1),f(s_2) such that Re(s_1) = Re(s_2) never fuse* into the same hair. 


Some questions are easier or harder than others. But ALL are intuitionistic imho.

These questions are both concrete and vague.

A) what superexponential is used for exp^[r] ?
B) what f(s) is used ?

But when we pick those 2 (A,B) the question is very concrete.

..and it makes sense to pick an exp^[r] based on the f(s) method.
( even though at present , even that might be a dispute or choice how to actually do it )

Yes this is very related and similar to a recent topic :

f(h(s)) = exp(f(s))

h(s) = g(exp(f(s))).

So much depends on h(s).
In fact it makes sense to define exp^[r] based on the f(s) resp h(s).

Since h(s) is close to s + 1 near the real line , the lenght argument makes sense.
And exp^[1/2](v) seems iso to ( approximate ) length increase of 1/2.

The term partition also relates here.

if an infinite set of dense hairs never intersect they locally partition a dense complex space.



Messages In This Thread
f(s),h(s) and "hairs". - by tommy1729 - 05/20/2021, 11:54 PM
RE: f(s),h(s) and "hairs". - by tommy1729 - 05/21/2021, 12:01 AM
RE: f(s),h(s) and "hairs". - by MphLee - 05/21/2021, 05:32 PM
RE: f(s),h(s) and "hairs". - by Gottfried - 05/21/2021, 07:23 PM
RE: f(s),h(s) and "hairs". - by tommy1729 - 05/22/2021, 12:24 PM
RE: f(s),h(s) and "hairs". - by JmsNxn - 05/23/2021, 12:42 AM

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