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Here i present the idea of continuous dynamics as alternating waves.

For instance take a starting point x for which we want to define an Abel function by
letting n go to oo :

G^[-n]( f^[n+z](x)) = A(z) + C

Where A(z) is the super for G.

However this requires that the iterates of f and g behave similar.

For instance the Julia set for f = x^2 + 1 ( g = sqrt ) does brake that condition.

Let the f iterates of a line segment On the complex plane containing x be red curves and Blue curves for g iterates.  Then those should make an alternating pattern of red and Blue waves.

So red Blue red Blue etc

Not red Blue blue red   !!

Now continu iterations may intersect like the folium of descartes

But as long as the pattern Of alternation holds that is ok.

This idea implies that the boetcher function for f = x^2 + 1 is analytic somewhere.
And also for my 2sinh method it implies analytic somewhere.

We can identify were the pattern of the alternating waves breaks down.

For instance for the case f = x^2  + 1. ;

x^ 4 = x^2 + 1

(x^ 2 + 1)^2 + 1 = x^2

...

Notice this is g(g(x)) = f(x) , showing f and g Grow at a locally different speed.

Similarly for f = 2sinh(x) and g(x) = exp(x).
[ basically my 2sinh method ]

Exp(x) = x

2Sinh(x) = x

Exp(exp(x)) = 2sinh(x)

2sinh(2sinh(x)) = exp(x)

...

Notice that these equations solve to an x with small real part or Large im part.

So the method seems to work near the Large positive reals.

Regards

Tommy1729

And

As main references.

The concept of growth is also related.

Slog ( f^[n] ) - n

Regards

Tommy1729
This resembles the idea of flows or vector calculus.
And ofcourse it also relates to the M-test.

Regards

Tommy1729