Second derivitive of a dynamical system

[Daniel]

The idea here is that the Taylor series for an arbitrary dynamical system \( f^t(x) \) can be derived. The second derivitive's dependancy on a geometrical progression results in the different types of fixed points. The higher derivitives also are built up from nested geometrical progressions. The regular formula for geometrical progressions breaks down when the roots of unity are involved which gives an explaination for why therre are different types of fixed points.