05/22/2023, 10:23 PM

Everybody who ever studied special relativity or hyperbolic trig knows this function

tanh(x)

It is itself connected to superfunctions because it is has an addition formula.

But what I want to focus on here is that for real 0 < A < 1

tanh(A x) has a nice attracting fixpoint at 0 and it is asymptotic to a positive constant (1) for large positive x.

It is also strictly increasing.

So this is very well suited to do complex dynamics on it !

We focus on the positive reals here.

We cannot get nice dynamics from iterating constant functions but we do get them here !

So we consider

tanh^[r](A x)

for r > 0.

Notice for every such r

lim tanh^[r](A x) = C_r

Where C is a constant depending on r.

This is all very interesting.

We get constants for 0 < r < 1 ; lim ( x to - oo ) exp^[r](x) as well but there we get that

1) we do not have a fixpoint,

2) it depends on our method used.

3) probably no hope for a good asymptotic with simpler functions.

But here it is probably nicer.

It also opens a new can of questions : How does dynamics behave for functions approaching a constant ?

What is the effect of the speed at which the function goes to a constant ?

We can slow the speed here by using

tanh(A x)^(2m+1)

for integer m >= 0.

Although this does not make power singularities, it does make the function flat at 0 for integer m > 0.

So maybe better to use

f(x,A,m) = ( tanh(A x) + tanh(A x)^(2m+1) )/2

Notice we are still talking about iterating periodic functions just like exp(x) is.

In particular I considered

tanh^[2^(-n)] (A x)

For visual reasons and algebra :

f_1(f_1(x)) = tanh(A x)

f_2(f_2(x)) = f_1(x)

f_3(f_3(x)) = f_2(x)

etc

half-iterates are easier visually , you get these nice " rectangles of iterations ".

Also tanh does not converge as fast as the gaussian erf.

The addition formula and other related ones might come in handy.

Im convinced that " starting simple is best " and starting simple (with iterating functions that go to a constant) ; implies starting with tanh(A x).

This is from the point of view of the positive reals.

By SYMMETRY we understand the situation for negative reals too.

I focus on reals here but the complex situation is also worth considering ofcourse.

tanh(A x) has many poles and fixpoints in the complex plane though, so that makes things harder.

Chaos also exists.

Unfortunately this implies taylor series are not easy here.

Keep in mind that is just one of those quick intro's ; many more ideas exist. And generalizations.

As usual.

regards

tommy1729

tanh(x)

It is itself connected to superfunctions because it is has an addition formula.

But what I want to focus on here is that for real 0 < A < 1

tanh(A x) has a nice attracting fixpoint at 0 and it is asymptotic to a positive constant (1) for large positive x.

It is also strictly increasing.

So this is very well suited to do complex dynamics on it !

We focus on the positive reals here.

We cannot get nice dynamics from iterating constant functions but we do get them here !

So we consider

tanh^[r](A x)

for r > 0.

Notice for every such r

lim tanh^[r](A x) = C_r

Where C is a constant depending on r.

This is all very interesting.

We get constants for 0 < r < 1 ; lim ( x to - oo ) exp^[r](x) as well but there we get that

1) we do not have a fixpoint,

2) it depends on our method used.

3) probably no hope for a good asymptotic with simpler functions.

But here it is probably nicer.

It also opens a new can of questions : How does dynamics behave for functions approaching a constant ?

What is the effect of the speed at which the function goes to a constant ?

We can slow the speed here by using

tanh(A x)^(2m+1)

for integer m >= 0.

Although this does not make power singularities, it does make the function flat at 0 for integer m > 0.

So maybe better to use

f(x,A,m) = ( tanh(A x) + tanh(A x)^(2m+1) )/2

Notice we are still talking about iterating periodic functions just like exp(x) is.

In particular I considered

tanh^[2^(-n)] (A x)

For visual reasons and algebra :

f_1(f_1(x)) = tanh(A x)

f_2(f_2(x)) = f_1(x)

f_3(f_3(x)) = f_2(x)

etc

half-iterates are easier visually , you get these nice " rectangles of iterations ".

Also tanh does not converge as fast as the gaussian erf.

The addition formula and other related ones might come in handy.

Im convinced that " starting simple is best " and starting simple (with iterating functions that go to a constant) ; implies starting with tanh(A x).

This is from the point of view of the positive reals.

By SYMMETRY we understand the situation for negative reals too.

I focus on reals here but the complex situation is also worth considering ofcourse.

tanh(A x) has many poles and fixpoints in the complex plane though, so that makes things harder.

Chaos also exists.

Unfortunately this implies taylor series are not easy here.

Keep in mind that is just one of those quick intro's ; many more ideas exist. And generalizations.

As usual.

regards

tommy1729