# 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)
* polynomial map
* repelling case: disconnected Julia set and only one basin ( superattractin with period 1: basin of infinity)
* 2 basins each with only one component
* 2 basins: exterior ( 1 component) and interior: consist of infinitely many components (attracting case)
* 2 basins: exterior and interior. Exterior consist of only one component ( superattracting with period 1). Interior consist of infinitely many components. Imediate basin of attraction consist of 3 components cointaining period 3 cycle
* Parabolic attractor belongs to Julia set
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,

## by attrators
### 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 :
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 $center=z_{c}r=z_{p}$

### Parabolic case: petal

* periods 1-2 and circle target
* fixed point belongs to Julia set and to the boundary of the trap ( here circle)
* periods greater than 2 so triangle targets
* triangle trap in parabolic case for t = 1/30
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
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 :
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
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
* parabolic case

### 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
* Basilica
* c = -1.24
* Douady Rabbit

### p-norm disk

See also
* kf
* DLD

# References

* ↑ Tessellation of the Interior of Filled Julia Sets by Tomoki Kawahira
* ↑ {Peitgen, H.O. and Fisher, Y. and Saupe, D. and McGuire, M. and Voss, R.F. and Barnsley, M.F. and Devaney, R.L. and Mandelbrot, B.B.}, (2012). The Science of Fractal Images. Springer Science & Business Media, 2012. p. 187. ISBN 9781461237846.{{cite book}}: CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)
* ↑ fractalforums.org : additionnal-bailout-variations-on-kalles-fraktaler