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Finding continue iteration cycles.

=> equivalent to solving

A(z)
F(z+1) = A(F(z))
G(s,z) = F(F^-1(z) + s)
Z=(a,b) = a + b i
Jordan curve C(a,b) given by D(a,b) = 0.
H(z) = G ' (0,z) ds - 1.

Arg H(z) = Tan C ' (a,b)
And H(z) =\= 0 on C(a,b).

Regards

Tommy1729
I edited the OP.

Regards

Tommy1729