Thefilled-in Julia set${\displaystyle K(f)}$ of a polynomial${\displaystyle f}$ is aJulia set and itsinterior,non-escaping set.

The filled-inJulia set${\displaystyle K(f)}$ of a polynomial${\displaystyle f}$ is defined as the set of all points${\displaystyle z}$ of the dynamical plane that haveboundedorbit with respect to${\displaystyle f}$$${\displaystyle K(f)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\left\{z\in \mathbb{C}:f^{(k)}(z)\nrightarrow \mathrm{\infty }\text{as}k\to \mathrm{\infty }\right\}}$$where:

• ${\displaystyle \mathbb{C}}$ is theset of complex numbers
• ${\displaystyle f^{(k)}(z)}$ is the${\displaystyle k}$ -foldcomposition of${\displaystyle f}$ with itself =iteration of function${\displaystyle f}$

The filled-in Julia set is the(absolute) complement of theattractive basin ofinfinity.$${\displaystyle K(f)=\mathbb{C}\setminus A_f(\mathrm{\infty })}$$

Theattractive basin ofinfinity is one of thecomponents of the Fatou set.$${\displaystyle A_f(\mathrm{\infty })=F_{\mathrm{\infty }}}$$

In other words, the filled-in Julia set is thecomplement of the unboundedFatou component:$${\displaystyle K(f)=F_{\mathrm{\infty }}^C.}$$

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TheJulia set is the commonboundary of the filled-in Julia set and theattractive basin ofinfinity$${\displaystyle J(f)=\mathrm{\partial }K(f)=\mathrm{\partial }{A}_f(\mathrm{\infty })}$$where:${\displaystyle A_f(\mathrm{\infty })}$ denotes theattractive basin ofinfinity = exterior of filled-in Julia set = set of escaping points for${\displaystyle f}$

$${\displaystyle A_f(\mathrm{\infty })\stackrel{\underset{\mathrm{d}\mathrm{e}\mathrm{f}}{}}{=}\{z\in \mathbb{C}:f^{(k)}(z)\to \mathrm{\infty }ask\to \mathrm{\infty }\}.}$$

If the filled-in Julia set has nointerior then theJulia set coincides with the filled-in Julia set. This happens when all the critical points of${\displaystyle f}$ are pre-periodic. Such critical points are often calledMisiurewicz points.

• 
  Rabbit Julia set with spine
• 
  Basilica Julia set with spine

The most studied polynomials are probablythose of the form${\displaystyle f(z)=z^2+c}$, which are often denoted by${\displaystyle f_c}$, where${\displaystyle c}$ is any complex number. In this case, the spine${\displaystyle S_c}$ of the filled Julia set${\displaystyle K}$ is defined asarc between${\displaystyle \beta }$-fixed point and${\displaystyle -\beta }$,$${\displaystyle S_c=\left[-\beta ,\beta \right]}$$with such properties:

• spine lies inside${\displaystyle K}$.^[1] This makes sense when${\displaystyle K}$ is connected and full^[2]
• spine is invariant under 180 degree rotation,
• spine is a finite topological tree,
• Critical point${\displaystyle z_{cr}=0}$ always belongs to the spine.^[3]
• ${\displaystyle \beta }$-fixed point is a landing point ofexternal ray of angle zero${\displaystyle {\mathcal{R}}_0^K}$,
• ${\displaystyle -\beta }$ is landing point ofexternal ray${\displaystyle {\mathcal{R}}_{1/2}^K}$.

Algorithms for constructing the spine:

• detailed version is described by A. Douady^[4]
• Simplified version of algorithm:
  • connect${\displaystyle -\beta }$ and${\displaystyle \beta }$ within${\displaystyle K}$ by an arc,
  • when${\displaystyle K}$ has empty interior then arc is unique,
  • otherwise take the shortest way that contains${\displaystyle 0}$.^[5]

Curve${\displaystyle R}$:$${\displaystyle R\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{R}_{1/2}\cup S_c\cup R_0}$$divides dynamical plane into two components.

• 
  Filled Julia set for f_c, c=1−φ=−0.618033988749…, where φ is theGolden ratio
• 
  Filled Julia with no interior = Julia set. It is for c=i.
• 
  Filled Julia set for c=−1+0.1*i. Here Julia set is the boundary of filled-in Julia set.
• 
  Douady rabbit
• 
  Filled Julia set for c = −0.8 + 0.156i.
• 
  Filled Julia set for c = 0.285 + 0.01i.
• 
  Filled Julia set for c = −1.476.

• airplane^[6]
• Douady rabbit
• dragon
• basilica orSan Marco fractal orSan Marco dragon
• cauliflower
• dendrite
• Siegel disc

1. Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986.ISBN 978-0-387-15851-8.
2. Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathematics Technical University of Denmark,MAT-Report no. 1996-42.

• Fractals
• Limit sets
• Complex dynamics

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