The filled-in Julia set
of a polynomial
is defined as the set of all points
of dynamical plane that have bounded orbit
with respect to
is set of complex numbers
is complex variable of function
is complex parameter of function
may be various functions. In typical case
is complex quadratic polynomial
is the -fold compositions of with itself = iteration of function
The filled-in Julia set is the (absolute) complement
of attractive basin
Attractive basin of infinity is one of components of the Fatou set.
In other words, the filled-in Julia set is the complement of the unbounded Fatou component:
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
denotes attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for
If filled-in Julia set has no interior then Julia set coincides with filled-in Julia set. It happens when is Misiurewicz point.
Spine of the filled Julia set is defined as arc between -fixed point and ,
with such properities:
- spine lays inside . This makes sense when is connected and full
- spine is invariant under 180 degree rotation,
- spine is a finite topological tree,
- Critical point always belongs to the spine.
- -fixed point is a landing point of external ray of angle zero ,
- is landing point of external ray .
Algorithms for constructiong the spine:
divides dynamical plane into 2 components.
- Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986. ISBN 978-0387158518.
- Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathemathics Technical University of Denmark , MAT-Report no. 1996-42