Filled Julia set

The filled-in Julia set   K ( f ) {displaystyle K(f)} of a polynomial   f {displaystyle f} is :Rabbit Julia set with spineBasilica Julia set with spineFilled Julia set for fc, c=φ−2=-0.38..., where φ means Golden ratioFilled 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 rabbitFilled Julia set for c = −0.4+0.6i.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. The filled-in Julia set   K ( f ) {displaystyle K(f)} of a polynomial   f {displaystyle f} is : 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 ) ↛ ∞   a s   k → ∞ } {displaystyle K(f) {overset {underset {mathrm {def} }{}}{=}} {zin mathbb {C} :f^{(k)}(z) ot o infty as k o infty }} where : C {displaystyle mathbb {C} } is the set of complex numbers   f ( k ) ( z ) {displaystyle f^{(k)}(z)} is the   k {displaystyle k} -fold composition of f {displaystyle f,} with itself = iteration of function f {displaystyle f,} 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} }{}}{=}} {zin mathbb {C} :f^{(k)}(z) o infty as k o infty }.}