05/20/2021, 11:54 PM

" 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 :

1)

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

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

2)

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.

regards

tommy1729

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 :

1)

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

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

2)

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.

regards

tommy1729