Cyclic dynamics f(-x) = T (f(x))

[tommy1729]

Let analytic a(z) = a^[r](z) for all z and some real r <> 1.

Cyclic dynamics fundamental equation.

F( a(z) ) = T ( F(z) )

With T entire.

Less general

By using g(q x) = a( g(x) ) , q = exp(2 pi i / r).

We get

W(q x) = T( W(x) ).

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Example

F(-x) = F(x) + pi i

Solution ln(x).

Can you find

F(-x) = g * F(x) + h

Note 1) we consider only nonconstant solutions
Note 2) we prefer a(x) to have exactly one fixpoint , so that its iterations are Unique !

Regards

Tommy1729

[tommy1729]

The solution seems to be constructible by using again a superfunction.

Example

F(-x) = e F(x)

Ln(-x) = ln(x) + pi i

Exp(x+1) = e exp(x)

So

F = exp(ln(x)/(pi i)) = x^(- 1/pi i)

Regards

Tommy1729

[tommy1729]

Assuming the assumption in the previous post was correct , we need to show that the equation keeps on being valid on the other branches too ... Or when that is not the case.

Analytic continuation probably is key here.

Its easy when the range and domain are on the same branch ( such as the example x^(- 1/pi i).

Then we get

A --> B --> A

I think this is not so hard.


Regards

Tommy1729