# 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,

## by attrators

### For infinite attractor

• Target set ${\displaystyle T\,}$  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 ${\displaystyle z=0\,}$  and radius =ER :
${\displaystyle T_{ER}=\{z:abs(z)>ER\}\,}$


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 :

## by the dynamics

Here

• ${\displaystyle z_{n}}$  is the last iteration of critical orbit
• ${\displaystyle center}$  is the center of the trap ( circle shape)
• ${\displaystyle z_{p}}$  is periodc / fixed point ( alfa fixed point)

Trap is the circle with center ${\displaystyle z=center}$  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

• ${\displaystyle z_{n}  and all points are inside Julia set
• ${\displaystyle AR=z_{p}-z_{n}}$
• 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 ${\displaystyle T\,}$  is an arbitrary set on dynamical plane containing infinity and not containing points of filled Julia set.

supperattracting case : here

• ${\displaystyle z_{cr}=z_{p}}$  so one have to set AR manually, like AR = 30*PixelWidth
• Stabilitu index = cabs(multiplier) = 0.0
• Center of attracting basin is ${\displaystyle center=z_{c}r=z_{p}}$

### Parabolic case: petal

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: ${\displaystyle z_{n}
• Stabilitu index cabs(multiplier) = 1.0
• here ${\displaystyle AR={\frac {z_{p}-z_{n}}{2}}}$

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: ${\displaystyle f^{-p}(z_{cr})}$ . 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(zp-z) < 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(zp-z) < 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(zp-z) < AR2 ){
return  i  % 255 ;
}

z = f(z);

}

return iColorOfUnknown;
}


## by the shape

### Exterior of circle

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

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

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

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

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


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

${\displaystyle T_{AR}=\{z:abs(z-z_{p})


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 ${\displaystyle s\,}$  centered at origin

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

### 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 [2]

### p-norm disk

2. {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)