Fractals/target set

=Definition=

Dynamical plane is divided into
 * Fatou set
 * Julisa set

Fatou set consist of one or more basins of attraction to the attractor.

Each basin of attraction has one or more critical points which fall into periodic obit ( attractor)

Target set
 * is a trap for forward orbit
 * is a set which captures any orbit tending to attractor (limit set = attracting cycle = fixed / periodic point).

=Types=

Criteria for classifications: one can divide it according to :
 * attractors ( finite or infinite)
 * dynamics ( hyperbolic, parabolic, elliptic )
 * shape ( bailout test)
 * destination
 * decomposition of target set: binary decomposition ( BDM), angular decomposition,

For infinite attractor

 * Target set $$T\,$$ is an arbitrary set on dynamical plane containing infinity and not containing points of Filled-in Fatou sets.
 * For escape time algorithms target set determines the shape of level sets and curves. It does not do it for other methods.
 * For escaping to infinity points ( basin of infinity = exterior of Julia set) it is  exterior of circle with center at origin $$z = 0 \,$$ and radius =ER :

$$T_{ER}=\{z:abs(z) > ER \} \,$$

Radius is named escape radius ( ER ) or bailout value. Radius should be greater than 2.

Infinity:
 * for polynomials infinity is superattracting fixed point. So in the exterior of Julia set (basin of attraction of infinity) the dynamics is the same for all polynomials. Escaping test ( = bailout test) can be used as a first universal tool.
 * for rational maps infinity is not a superattrating fixed point. It may be periodic point or not.

For finite attractors


For finite attractors see: target set by basin

See :
 * Internal Level Sets
 * Binary decomposition
 * Tessellation of the Interior of Filled Julia Sets by Tomoki KAWAHIRA

by the dynamics


Here
 * $$ z_n $$ is the last iteration of critical orbit
 * $$ center $$ is the center of the trap ( circle shape)
 * $$z_p$$ is periodc / fixed point ( alfa fixed point)

Trap is the circle with center $$ z = center $$ and  radius = AR

repeling case

 * Stability index = cabs(multiplier) > 1.0
 * periodc / fixed point ( alfa fixed point) is repelling = ther is no interior of Julia set

attracting but not supperattracting case

 * $$ z_n < z_p < z_p + AR $$ and all points are inside Julia set
 * $$ AR = z_p - z_n$$
 * Stabilitu index: 0.0 < cabs(multiplier) < 0.0

Elliptic case
For the elliptic dynamics, when there is a Siegel disc, the target set is an inner circle

Supperattracting case
Attractors:
 * Infinity is allways superattracting for forward iteration of polynomials. Target set here is an exterior of any shape containing all point of Julia set (and its interior)
 * finite asttractors can also be superattracting, when parameter c is a center ( nucleus) of hyperbolic component of Mandelbrot set

In case of forward iteration target set $$T\,$$ is an arbitrary set on dynamical plane containing infinity and not containing points of filled Julia set.

supperattracting case : here
 * $$ z_{cr} = z_p$$ so one have to set AR manually, like AR = 30*PixelWidth
 * Stabilitu index = cabs(multiplier) = 0.0
 * Center of attracting basin is $$ center = z_cr = z_p$$

Parabolic case: petal
In parabolic case trap can be for
 * drawing components of Julia set
 * BDM = parabolic checkerboard

In the parabolic case target set should be inside the petal

parabolic case for child period 1 and 2 the target set can have circle shape :
 * one should:
 * compute AR
 * change trap center to midpoint between attracting fixed point zp and the last iteration of critical orbit zn to get: $$ z_n < center < z_p$$
 * Stabilitu index cabs(multiplier) = 1.0
 * here $$ AR = \frac{z_p - z_n}{2}$$

Fof child periods > 2 petal can be triangle fragment of the circle around fixed point for the parent period.

by destination
It is important for parabolic case:
 * for Fatou basin ( color depends on the target set): circle around fixed point = trap for interior
 * for component of Fatou basin ( color proportional to to iteration modulo period) - triangle fragment of above circle = biggest triangle (zp, zprep, -zprep) = trap for components
 * for level set of Fatou basin ( color proportional to last iteration number ) = trap for components
 * for BDM or parabolic checkerboard : 2 smaller triangles (zp, zprecr, zcr) and (zp, zcr, -zprecr) = traps for BD

-zprecr zf     	    zcr zprecr

where
 * p is a period
 * zf = fixed point ( here period = 1)
 * zcr = critical point z=0
 * zprecr = precritical point = preimage of critical point: $$f^{-p}( z_{cr} )$$. Note that inverse function is multivalued so one should choose the proper preimage

by the shape

 * circle
 * square
 * Julia set
 * p-norm disk

Exterior of circle
This is typical target set. It is exterior of circle with center at origin $$z = 0 \,$$ and radius =ER:

$$T_{ER}=\{z:abs(z) > ER \} \,$$

Radius is named escape radius (ER) or bailout value.

Circle of radius=ER centered at the origin is: $$ \{z:abs(z) = ER \} \,$$

For escaping to infinity points ( basin of infinity = exterior of Julia set) it is  exterior of circle with center at origin $$z = 0 \,$$ and radius =ER :

$$T_{ER}=\{z:abs(z) > ER \} \,$$

Radius is named escape radius ( ER ) or bailout value. Radius should be greater than 2.

For finite attractors it is interior of the circle with center at periodic point

$$T_{AR}=\{z :abs(z - z_p) < AR \} \,$$

For parabolic periodic points
 * it is called a petal
 * petal is interior of the circle
 * center of petal circle is equal to midpoint between lat iteration and parabolic periodic point
 * parabolic periodic point belongs to Julia set

Exterior of square
Here target set is exterior of square of side length $$s\, $$ centered at origin

$$T_s=\{z: abs(re(z)) > s \mbox{or}  abs(im(z))>s \} \,$$

Julia sets
Escher like tilings is a modification of the level set method ( LSM/J). Here Level sets of escape time are different because targest set is different. Here target set is a scalled filled-in Julia set.

For more description see
 * Fractint : escher_julia
 * page 187 from The Science of fractal images by Heinz-Otto Peitgen, Dietmar Saupe, Springer

p-norm disk
See also
 * kf
 * DLD

=References=