The filled-in Julia set K ( f ) {\displaystyle K(f)} of a polynomial f {\displaystyle f} is a Julia set and its interior, non-escaping set.
The filled-in Julia set K ( f ) {\displaystyle K(f)} of a polynomial f {\displaystyle f} is defined as the set of all points z {\displaystyle z} of the dynamical plane that have bounded orbit with respect to f {\displaystyle f} K ( f ) = d e f { z ∈ ∈ --> C : f ( k ) ( z ) ↛ ∞ ∞ --> as k → → --> ∞ ∞ --> } {\displaystyle K(f){\overset {\mathrm {def} }{{}={}}}\left\{z\in \mathbb {C} :f^{(k)}(z)\not \to \infty ~{\text{as}}~k\to \infty \right\}} where:
The filled-in Julia set is the (absolute) complement of the attractive basin of infinity. K ( f ) = C ∖ ∖ --> A f ( ∞ ∞ --> ) {\displaystyle K(f)=\mathbb {C} \setminus A_{f}(\infty )}
The attractive basin of infinity is one of the components of the Fatou set. A f ( ∞ ∞ --> ) = F ∞ ∞ --> {\displaystyle A_{f}(\infty )=F_{\infty }}
In other words, the filled-in Julia set is the complement of the unbounded Fatou component: K ( f ) = F ∞ ∞ --> C . {\displaystyle K(f)=F_{\infty }^{C}.}
The Julia set is the common boundary of the filled-in Julia set and the attractive basin of infinity J ( f ) = ∂ ∂ --> K ( f ) = ∂ ∂ --> A f ( ∞ ∞ --> ) {\displaystyle J(f)=\partial K(f)=\partial A_{f}(\infty )} where: A f ( ∞ ∞ --> ) {\displaystyle A_{f}(\infty )} denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for f {\displaystyle f}
A f ( ∞ ∞ --> ) = d e f { z ∈ ∈ --> C : f ( k ) ( z ) → → --> ∞ ∞ --> a s k → → --> ∞ ∞ --> } . {\displaystyle A_{f}(\infty )\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{z\in \mathbb {C} :f^{(k)}(z)\to \infty \ as\ k\to \infty \}.}
If the filled-in Julia set has no interior then the Julia set coincides with the filled-in Julia set. This happens when all the critical points of f {\displaystyle f} are pre-periodic. Such critical points are often called Misiurewicz points.
The most studied polynomials are probably those of the form f ( z ) = z 2 + c {\displaystyle f(z)=z^{2}+c} , which are often denoted by f c {\displaystyle f_{c}} , where c {\displaystyle c} is any complex number. In this case, the spine S c {\displaystyle S_{c}} of the filled Julia set K {\displaystyle K} is defined as arc between β β --> {\displaystyle \beta } -fixed point and − − --> β β --> {\displaystyle -\beta } , S c = [ − − --> β β --> , β β --> ] {\displaystyle S_{c}=\left[-\beta ,\beta \right]} with such properties:
Algorithms for constructing the spine:
Curve R {\displaystyle R} : R = d e f R 1 / 2 ∪ ∪ --> S c ∪ ∪ --> R 0 {\displaystyle R{\overset {\mathrm {def} }{{}={}}}R_{1/2}\cup S_{c}\cup R_{0}} divides dynamical plane into two components.
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