08/24/2015, 11:56 PM

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) ).

---

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

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) ).

---

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