Fractals/Iterations in the complex plane/julia/interior

" Stable orbits of polynomials may :

  • converge to a (super)-attracting fixed point,
  • (coverge) to a parabolic fixed point (where the multiplier is a root of unity),
  • belong to a rotation domain (a simply connected domain on which the dynamics is conjugate to a rotation)." Lasse Rempe-Gillen [1]


Local discrete complex dynamics edit

Types of dynamics
  • attracting : hyperbolic dynamics
    • superattracting : the very fast ( = exponential) convergence to periodic cycle ( fixed point )
  • parabolic component = slow ( lazy ) dynamics = slow ( exponential slowdown) convergence to parabolic fixed point ( periodic cycle)
  • Siegel disc component = rotation around fixed point and never reach the fixed point

When Julia set is disconnected ther is no interior of Julia set ( critical fixed point is repelling ( or attracting to infinity)

References edit

  1. mathoverflow : attractive-basins-and-loops-in-julia-sets