Fractals/target set
Definition
editDynamical 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)

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
editCriteria 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
editFor infinite attractor
edit Target set is an arbitrary set on dynamical plane containing infinity and not containing points of Filledin 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 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
edit
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 ^{[1]}
by the dynamics
editHere
 is the last iteration of critical orbit
 is the center of the trap ( circle shape)
 is periodc / fixed point ( alfa fixed point)
Trap is the circle with center and radius = AR
repeling case
edit 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
edit and all points are inside Julia set
 Stabilitu index: 0.0 < cabs(multiplier) < 0.0
Elliptic case
editFor the elliptic dynamics, when there is a Siegel disc, the target set is an inner circle
Supperattracting case
editAttractors:
 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 is an arbitrary set on dynamical plane containing infinity and not containing points of filled Julia set.
supperattracting case : here
 so one have to set AR manually, like AR = 30*PixelWidth
 Stabilitu index = cabs(multiplier) = 0.0
 Center of attracting basin is
Parabolic case: petal
edit
fixed point belongs to Julia set and to the boundary of the trap ( here circle)

triangle trap in parabolic case for t = 1/30
In parabolic case trap can be for
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:
 Stabilitu index cabs(multiplier) = 1.0
 here
Fof child periods > 2 petal can be triangle fragment of the circle around fixed point for the parent period.
by destination
editIt 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: . Note that inverse function is multivalued so one should choose the proper preimage
unsigned char ComputeColorOfFatouBasins (complex double z)
{
int i; // number of iteration
for (i = 0; i < IterMax; ++i)
{
// infinity is superattracting here !!!!!
if ( cabs2(z) > ER2 ){ return iColorOfExterior;}
// 1 Attraction basins
if ( cabs2(zpz) < AR2 ){ return iColorOfInterior;}
z = f(z); // iteration: z(n+1) = f(zn)
}
return iColorOfUnknown;
}
unsigned char ComputeColorOfFatouComponents (complex double z)
{
int i; // number of iteration
for (i = 0; i < IterMax; ++i)
{
// infinity is superattracting here !!!!!
if ( cabs2(z) > ER2 ){ return iColorOfExterior;}
//1 Attraction basins
if ( cabs2(zpz) < AR2 )
{ return iColorOfBasin1  (i % period)*20;} // number of components in imediate basin = period
z = f(z); // iteration: z(n+1) = f(zn)
}
return iColorOfUnknown;
}
unsigned char ComputeColorOfLSM (complex double z)
{
//double r2;
int i; // number of iteration
for (i = 0; i < IterMax_LSM; ++i)
{
if ( cabs2(z) > ER2 ){ return iColorOfExterior;}
//1 Attraction basins
if ( cabs2(zpz) < AR2 ){
return i % 255 ;
}
z = f(z);
}
return iColorOfUnknown;
}
by the shape
edit circle
 square
 Julia set
 pnorm disk
Exterior of circle
editThis is typical target set. It is exterior of circle with center at origin and radius =ER:
Radius is named escape radius (ER) or bailout value.
Circle of radius=ER centered at the origin is:
For escaping to infinity points ( basin of infinity = exterior of Julia set) it is exterior of circle with center at origin 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
editHere target set is exterior of square of side length centered at origin
Julia sets
editEscher 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 filledin Julia set.
For more description see
 Fractint : escher_julia
 page 187 from The Science of fractal images by HeinzOtto Peitgen, Dietmar Saupe, Springer ^{[2]}

Basilica

c = 1.24

Douady Rabbit
pnorm disk
editSee also
References
edit ↑ 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}}
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