Periodic Points of Complex Quadratic Mappings - Stability of Periodic Points (orbit) - Multiplier

The multiplier ( or eigenvalue, derivative ) of rational map at fixed point is defined as :

m(f,z_0)=\lambda = \begin{cases} f_c'(z_0), &\mbox{if }z_0\ne \infty \\ \frac{1}{f_c'(z_0)}, & \mbox{if }z_0 = \infty \end{cases}

where is first derivative of with respect to at .

Because the multiplier is the same at all periodic points, it can be called a multiplier of periodic orbit.

Multiplier is:

  • complex number,
  • invariant under conjugation of any rational map at its fixed point
  • used to check stability of periodic (also fixed) points with stability index :

Periodic point is :

  • attracting when
    • super-attracting when
    • attracting but not super-attracting when
  • indifferent when
    • rationally indifferent or parabolic if is a root of unity
    • irrationally indifferent if but multiplier is not a root of unity
  • repelling when

Where do periodic points belong?

  • attracting is always in Fatou set
  • repelling is in the Julia set
  • Indifferent fixed points may be in the one or in the other. Parabolic periodic point is in Julia set.

Read more about this topic:  Periodic Points Of Complex Quadratic Mappings, Stability of Periodic Points (orbit)