Ah now I see what you meant by "from -1", namely that 0 is mapped to -1 by the superfunction. As indicated above we just write where unfortunately the should be a \mapsto having a vertical bar at the left, which is however not shown in this TeX-derivate.

Note that regular superfunctions are determined up to translations along the x-Axis in this case .

Instead of we can put an arbitrary different constant, via which we can choose initial conditions different from .

For example would be reached by and by , etc.

Regarding the iteration of quadratic polynomials there is also a very interesting article about the impossibility to do so in the whole complex plane.

[1] R. E. Rice, B. Schweizer, and A. Sklar. When is f(f(z))=az^2 + bz + c? Am.Math.Mon., 87 : 252 −−263,1980.

Not only the impossibility to have analytic, or continous halfiterates; no, there are no halfiterates (functions on ) at all!

Note that regular superfunctions are determined up to translations along the x-Axis in this case .

Instead of we can put an arbitrary different constant, via which we can choose initial conditions different from .

For example would be reached by and by , etc.

Regarding the iteration of quadratic polynomials there is also a very interesting article about the impossibility to do so in the whole complex plane.

[1] R. E. Rice, B. Schweizer, and A. Sklar. When is f(f(z))=az^2 + bz + c? Am.Math.Mon., 87 : 252 −−263,1980.

Not only the impossibility to have analytic, or continous halfiterates; no, there are no halfiterates (functions on ) at all!