Definitions

Filled Julia set

Filled Julia set

The filled-in Julia set K(f_c) of a polynomial f _c is defined as the set of all points z, of dynamical plane that have bounded orbit with respect to f _c

K(f_c) overset{underset{mathrm{def}}{}}{=} { z in mathbb{C} : f^{(k)} _c (z) notto infty as k to infty }
where :

mathbb{C} is set of complex numbers

z, is complex variable of function f _c (z)
c, is complex parameter of function f _c (z)

f_c:mathbb Ctomathbb C
f_c may be various functions. In typical case f_c is complex quadratic polynomial.

f^{(k)} _c (z) is the k -fold compositions of f _c, with itself = iteration of function f _c,

Relation to the Fatou set

The filled-in Julia set is the (absolute) complement of attractive basin of infinity.
K(f_c) = mathbb{C} setminus A_{f_c}(infty)

Attractive basin of infinity is one of components of the Fatou set.
A_{f_c}(infty) = F_infty

In other words, the filled-in Julia set is the complement of the unbounded Fatou component:
K(f_c) = F_infty^C

Relation between Julia, filled-in Julia set and attractive basin of infinity

Julia set is common boundary of filled-in Julia set and attractive basin of infinity

J(f_c), = partial K(f_c) =partial A_{f_c}(infty)

where :
A_{f_c}(infty) denotes attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for f_c

A_{f_c}(infty) overset{underset{mathrm{def}}{}}{=} { z in mathbb{C} : f^{(k)} _c (z) to infty as k to infty }


If filled-in Julia set has no interior then Julia set coincides with filled-in Julia set. It happens when c, is Misiurewicz point.

Spine

Spine S_c, of the filled Julia set K , is defined as arc between beta, -fixed point and -beta,,

S_c = left [- beta , beta right ],

with such properities:

  • spine lays inside K ,. This makes sense when K, is connected and full
  • spine is invariant under 180 degree rotation,
  • spine is a finite topological tree,
  • Critical point z_{cr} = 0 , always belongs to the spine.
  • beta, -fixed point is a landing point of external ray of angle zero mathcal{R}^K _0,
  • -beta, is landing point of external ray mathcal{R}^K _{1/2}.

Algorithms for constructiong the spine:

Curve R, :

R overset{underset{mathrm{def}}{}}{=} R_{1/2} cup S_c cup R_0 ,

divides dynamical plane into 2 components.

Images

References

  1. Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986. ISBN 978-0387158518.
  2. Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathemathics Technical University of Denmark , MAT-Report no. 1996-42

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