Julia crowd

The Julia sets, first described by Gaston Maurice Julia and Pierre Fatou , are subsets of the complex number plane , with a Julia set belonging to every holomorphic or meromorphic function. Often the Julia sets are fractal sets. The complement of the Julia set is called the Fatou set.

Julia set (white line) of a quadratic polynomial. The dark Fatou crowd is shaded green or purple.

If you apply a completely defined function over and over again to its function values, then a sequence of complex numbers results for each : ${\displaystyle \mathbb {C} }$ ${\displaystyle f}$ ${\displaystyle z}$

${\displaystyle z\mapsto f(z)\mapsto f(f(z))\mapsto \cdots }$

Depending on the start value , this sequence can show two fundamentally different behaviors: ${\displaystyle z}$

• A small change in the starting value leads to practically the same result, the dynamics are stable in a certain sense: the starting value is assigned to the Fatou set.
• Even the smallest change in the start value leads to a completely different behavior of the sequence, the dynamics depend “chaotically” on the start value: The start value belongs to the Julia set.

background

The Newton's method is one of the best known and most widely used method for solution of nonlinear equations. You have the equation to be solved in the form

${\displaystyle p(z)\;{\stackrel {.}{=}}\;0}$

then zeros of a function are to be found. If the function is differentiable , Newton's method transforms the static problem into a dynamic process: It delivers an iteration rule of the shape ${\displaystyle z}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p(z)=0}$

${\displaystyle z_{n+1}=f(z_{n})\quad {\text{ mit }}\quad f(z)=z-{\frac {p(z)}{p'(z)}}}$

with the following properties:

• The zeros of become fixed points of ${\displaystyle p}$ ${\displaystyle f.}$
• If the initial value of the iteration close to a zero of then converges the Newton iteration against the belonging of fixed point and thus against this root. ${\displaystyle z_{0}}$ ${\displaystyle p,}$ ${\displaystyle f}$

So you just have to have an approximate solution to the problem. The fixed points act in a similar way to the centers of force fields that attract particles in their vicinity. With each iteration step, the particles move closer to the power source.

In terms of its conception, Newton's method is - like other fixed point iterations - a local method whose behavior is known when one is close to a zero. But what happens if we move further away from the points of attraction, and what are the boundaries between the catchment areas of the individual power sources?

Julia set of the Newton iteration (the Newton fractal ) of a cubic polynomial:
The turquoise , beige and pink colored areas are the three catchment areas of the three roots of the polynomial.
Starting values ​​from the red areas are attracted to a cycle of length 2, so they do not converge towards one of the zeros. Starting values ​​from the white areas, which form the border between the individual catchment areas, hop around wildly and also do not converge towards one of the zeros.

Serious studies of the global dynamics of the method date back to 1879 when Lord Arthur Cayley extended the problem from real numbers to complex numbers and proposed global studies:

“In connexion herewith, throwing aside the restrictions as to reality, we have what I call the Newton-Fourier Imaginary Problem. [...] The problem is to determine the regions of the plane, such that P being taken at pleasure anywhere within one region we arrive ultimately at the point A; anywhere within another region at the point B; and so for the several points representing the roots of the equation. "

- Arthur Cayley

In doing so, however , he encountered insurmountable problems in the event that a polynomial is third degree, so that he finally stopped his investigations: ${\displaystyle p}$

"The solution is easy and elegant in the case of a quadratic equation, but the next succeeding case of the cubic equation appears to present considerable difficulty."

- Arthur Cayley

Against this background, the French Pierre Fatou and Gaston Julia developed their theory of the iterations of rational functions in the complex plane at the beginning of the 20th century , that is, the theory of discrete dynamic systems of form

${\displaystyle z\mapsto f(z)}$

with a meromorphic function ${\displaystyle f.}$

properties

Is thus a meromorphic function on the completion of the complex numbers, that is the quotient of a holomorphic function and a polynomial (whose common zero points are already reduced, for. Example, the quotient of two relatively prime polynomials, or a sine function by a polynomial, wherein zeros be reduced to integer multiples). In addition, let the degree of greater than The degree of a meromorphic function is the maximum of the degrees of the relatively prime polynomials in the numerator and denominator. The degree generally indicates how many archetypes a point has. Depending on the dynamics of the process for a certain start value, this value is assigned to one of two quantities: ${\displaystyle f}$ ${\displaystyle \pi }$ ${\displaystyle f}$ ${\displaystyle 1.}$ ${\displaystyle z\mapsto f(z)}$

Fatou crowd

The starting values ​​from this set lead to constant dynamics under iteration , that is: if the starting value changes only a little, then the dynamics also show a similar behavior.

Julia crowd

The points in this set lead to unstable processes: every change in the start value, no matter how small, leads to a completely different dynamic.

The number ball is the disjoint union of these two sets. So each point belongs to either the Fatou set or the Julia set. The Julia set of a function is denoted as and the Fatou set as ${\displaystyle J_{f}}$ ${\displaystyle F_{f}.}$

The historical definition of the Julia set, as it comes from Fatou and Julia and can be read below, is neither particularly intuitive nor descriptive. Therefore, some properties of these sets are summarized here, for which a few basic terms are needed first.

Terms

For each value defines the recursion ${\displaystyle z_{0}\in {\overline {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}}$

${\displaystyle z_{n+1}=f(z_{n}){\text{ mit }}n\in \mathbb {N} _{0}{\text{ und einem Startwert }}z_{0}}$

a sequence of points on the Riemann number sphere. This sequence is also known as the orbit of : ${\displaystyle z_{0}}$

${\displaystyle \operatorname {Or} ^{+}(z_{0})=\{f^{n}(z_{0})\}_{n\in \mathbb {N} _{0}}}$

${\displaystyle f^{n}}$always means repeated execution of and should not be confused with the -th power. The definition of the inverse orbit is a little different because it is generally not clearly reversible. The inverse orbit of a point consists of all points that will be mapped onto it at some point: ${\displaystyle n}$ ${\displaystyle f}$ ${\displaystyle n}$ ${\displaystyle f}$ ${\displaystyle z_{0}}$

${\displaystyle \operatorname {Or} ^{-}(z_{0})=\{z\in {\overline {\mathbb {C} }}:f^{k}(z)=z_{0}{\text{ für ein }}k\in \mathbb {N} _{0}\}}$

If for one , then it is called a periodic point and the orbit ${\displaystyle z_{n}=z_{0}}$ ${\displaystyle n}$ ${\displaystyle z_{0}}$

${\displaystyle \gamma =\{z_{0},f(z_{0}),f^{2}(z_{0}),\ldots ,f^{n-1}(z_{0})\}}$

is called periodic orbit or cycle. If the smallest natural number with this property is called the period of the cycle. If this is true, then a fixed point of Evidently is a periodic point whose period is the same , a fixed point of On the basis of the derivative , one can characterize the stability of a periodic point. Be to it ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n=1}$ ${\displaystyle f(z_{0})=z_{0}}$ ${\displaystyle z_{0}}$ ${\displaystyle f.}$ ${\displaystyle f,}$ ${\displaystyle n}$ ${\displaystyle f^{n}.}$

${\displaystyle \lambda =\left(f^{n}\right)'(z_{0}).}$

Then the periodic point is called

• strongly attractive if ${\displaystyle \lambda =0,}$
• attractive if ${\displaystyle 0<|\lambda |<1,}$
• indifferent if ${\displaystyle |\lambda |=1,}$
• repulsive when ${\displaystyle |\lambda |>1.}$

By applying the chain rule one can see that for all points of the cycle has the same value, and analogously this cycle is then called (strongly) attractive, indifferent or repulsive. ${\displaystyle (f^{n})'}$

This naming is motivated by the following observation: In the case, it behaves in the vicinity of the fixed point in the same way as in an environment of zero. During iteration, values ​​therefore move closer and closer to the fixed point, if applies, and for the values ​​move further and further away from the fixed point. Under the iteration takes the checkpoint so values in its vicinity on or he pushes it off. The case for is more complicated, and for the values ​​are attracted at least as strongly as in an environment of ${\displaystyle |\lambda |\neq 0,1}$ ${\displaystyle f^{n}}$ ${\displaystyle z_{0}}$ ${\displaystyle g(z)=\lambda z}$ ${\displaystyle |\lambda |<1}$ ${\displaystyle |\lambda |>1}$ ${\displaystyle |\lambda |=1}$ ${\displaystyle \lambda =0}$ ${\displaystyle g(z)=a\cdot z^{2}}$ ${\displaystyle 0.}$

Is an attractive fixed point of then the amount is called ${\displaystyle z_{0}}$ ${\displaystyle f,}$

${\displaystyle A(z_{0})=\left\{z\in {\overline {\mathbb {C} }}:\lim _{k\to \infty }f^{k}(z)=z_{0}\right\}}$

the catchment area of the fixed point. The set thus consists of all points whose orbit converges to. Apparently this set contains the inverse orbit of Das comes from the English basin of attraction (catchment area / collecting basin of the attractor, in this case collecting basin of an attractive fixed point or cycle). If the period is an attractive periodic cycle , then each of the fixed points has its catchment area, and denotes the union of these catchment areas. ${\displaystyle A(z_{0})}$ ${\displaystyle z_{0}}$ ${\displaystyle z_{0}.}$ ${\displaystyle A}$ ${\displaystyle \gamma }$ ${\displaystyle n}$ ${\displaystyle f^{k}(z_{0}),\ k=0,\dotsc ,n-1}$ ${\displaystyle A(\gamma )}$

definition

A possible definition of the Julia set occurs via the set of its repulsive periodic points:

${\displaystyle J_{f}\;{\overset {\mathrm {def} }{=}}\;\operatorname {Abschluss} \left\{z\in {\overline {\mathbb {C} }}:z{\text{ ist abstoßender periodischer Punkt von }}f\right\},}$

where "closure" means the topological closure . This is the definition on which Julia based his theory. The starting point of Fatouschen's work was another definition given below.

Each element of the Julia set can thus be represented as the limit value of a convergent sequence that consists only of repulsive periodic points of . ${\displaystyle f}$

Basic properties

Some properties of the Julia set are:

1. The set of repulsive periodic points is dense in ${\displaystyle J_{f}.}$
2. ${\displaystyle J_{f}\neq \emptyset }$and contains uncountably many points.
3. ${\displaystyle f(J_{f})\,\!=J_{f}=f^{-1}(J_{f})}$
4. The Julia sets of and are identical. ${\displaystyle f}$ ${\displaystyle f^{k},\ k=1,2,\ldots }$
5. For each out , the inverse orbit is dense in ${\displaystyle z}$ ${\displaystyle J_{f}}$ ${\displaystyle \operatorname {Or} ^{-}(z)}$ ${\displaystyle J_{f}.}$
6. If an attractive cycle is then for the catchment area of ​​the cycle and its edge : and ${\displaystyle \gamma }$ ${\displaystyle f,}$ ${\displaystyle A(\gamma )\subset F_{f}={\overline {\mathbb {C} }}\setminus J_{f}}$ ${\displaystyle \partial A(\gamma )=J_{f}.}$
7. Let be an element of the Julia set and a neighborhood of Then there is a natural number with ${\displaystyle z}$ ${\displaystyle U}$ ${\displaystyle z.}$ ${\displaystyle n=n(U)}$ ${\displaystyle f^{n}(J_{f}\cap U)=J_{f}.}$
8. If the Julia set has interior points , then we have ${\displaystyle J_{f}={\overline {\mathbb {C} }}.}$

Explanations

1. This follows directly from the definition given.
2. Every rational function has a considerable supply of repulsive periodic points.
3. The Julia set is invariant if one applies pointwise to the Julia set, the result is again the Julia set. The same applies to the number of archetypes. ${\displaystyle f:}$ ${\displaystyle f}$
4. Follows by induction from the previous point.
5. This point inspires a method for visualizing the Julia set by backward iteration. However, the archetypes are not evenly distributed in and the archetypes are generally not easy to determine. ${\displaystyle J_{f},}$
6. This property can be used for an imaging process if one knows an attractive cycle. If a point is in the catchment area of ​​this cycle, it is colored white, for example, otherwise black. The Julia set is then the boundary between the two areas. Also, this property says that the Julia set must have fractal properties in many cases . Does the function z. B. more than two attractive fixed points then that means that each point of the Julia set lies on the edge of each catchment area; and all catchment areas have the same edge. ${\displaystyle f}$ ${\displaystyle a,b,c,\dotsc ,}$
       ${\displaystyle \partial A(a)=J_{f}=\partial A(b)=J_{f}=\partial A(c)=\cdots ,}$
   
7. The Julia set can be reconstructed from any small piece by applying the function to it finitely often (point by point). In addition, the Julia set has no isolated points . ${\displaystyle f}$

Critical points

A point is called the critical point of , if in any environment of is reversible. If differentiable , then a critical point is through ${\displaystyle z}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle z}$ ${\displaystyle f}$

${\displaystyle f'\!\,(z)=0}$

characterized. In every catchment area that belongs to a (strongly) attractive attractor, there is at least one critical point. By considering the critical points of a function, statements can be made about the dynamics of this function.

A well-known example of this is the Mandelbrot set , whose relation to certain Julia sets is explained below. The Mandelbrot set maps the different behavior of the critical point of the mapping for different values ​​of ${\displaystyle 0}$ ${\displaystyle z\mapsto z^{2}+c}$ ${\displaystyle c.}$

Julia sets of polynomials

Julia sets of the function for different parameters . This is under each graphic. ${\displaystyle z\mapsto z^{2}+c}$ ${\displaystyle c}$

A simple way to define the Julia set of a polynomial is by means of recursion ${\displaystyle p}$

${\displaystyle z_{n+1}=p(z_{n})}$

with a starting value ${\displaystyle z_{0}.}$

The set is defined as the set of all complex numbers whose absolute value remains limited after any number of iteration steps. The Julia set is then the edge of this set. is referred to as the completed Julia set or occasionally imprecisely as the Julia set itself. One can show that is limited. ${\displaystyle K_{p}}$ ${\displaystyle z_{0},}$ ${\displaystyle J_{p}}$ ${\displaystyle K_{p}}$ ${\displaystyle K_{p}}$

This definition is the direct implementation of property 6: For a polynomial there is an attractive fixed point. The Julia set thus results as the edge of the catchment area of ​​this fixed point. If a point lies in it, then it finally converges to or - if the standard metric is used - its amount grows across all boundaries. If its amount remains limited, then it belongs to the catchment area of ​​another attractor or to the Julia set itself. ${\displaystyle \infty }$ ${\displaystyle \infty }$

This definition is usually used to generate graphics because it can be easily translated into a computer program.

The same definition can be used for meromorphic functions whose numerator degree is at least greater than their denominator degree, since there is also an attractive fixed point for such functions . ${\displaystyle 2}$ ${\displaystyle \infty }$

Dynamics using the example ${\displaystyle f(z)=z^{2}}$

The Julia set for is the edge of the unit circle. ${\displaystyle z\mapsto z^{2}+0}$

With this simple example, many properties of the Julia set can be demonstrated.

The function has three fixed points: The following applies for these points Since the derivative vanishes in and in , these two fixed points are attractive fixed points, while is repulsive. All starting values ​​whose absolute value is less than converge to and all starting values ​​whose absolute value is greater than converge toward ${\displaystyle f(z)=z^{2}}$ ${\displaystyle 0,1,\infty .}$ ${\displaystyle f(z)=z.}$ ${\displaystyle 0}$ ${\displaystyle \infty }$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle \infty :}$

${\displaystyle A(\infty )=\{z\in {\overline {\mathbb {C} }}:|z|>1\}}$

${\displaystyle A(0)\,\,\,=\{z\in {\overline {\mathbb {C} }}:|z|<1\}}$

In the remaining case lies on the unit circle , has the representation and the application of doubled, i.e. only the (real) exponent in the polar coordinate representation, the absolute value of the number always remains the same . The exponent can always be chosen so that it lies in the half-open interval . If one only considers the effect of on the variable in the exponent, then corresponds to the figure ${\displaystyle |z|=1}$ ${\displaystyle z}$ ${\displaystyle z=e^{2\pi i\cdot x}}$ ${\displaystyle f(z)=z^{2}=e^{2\pi i\cdot 2x}.}$ ${\displaystyle f}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle [0,1)}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle f}$

${\displaystyle g\colon x\mapsto 2x\mod 1}$

on the real interval, i.e. a multiplication by , whereby only the decimal places are relevant. The fixed point of becomes the fixed point of . If you iterate the value with , the result is the result ${\displaystyle [0,1),}$ ${\displaystyle 2}$ ${\displaystyle 1}$ ${\displaystyle f}$ ${\displaystyle 0}$ ${\displaystyle g}$ ${\displaystyle 1/3}$ ${\displaystyle g}$

${\displaystyle {\tfrac {1}{3}}\mapsto {\tfrac {2}{3}}\mapsto {\tfrac {4}{3}}\equiv {\tfrac {1}{3}}\mapsto {\tfrac {2}{3}}\mapsto \cdots }$

So is a periodic point, as in the representation of a number as a dual fraction be multiplying with only the digits shifted one place to the left, and the decimal point is always the "mod" to set such an example can be seen: ${\displaystyle 1/3}$ ${\displaystyle 2/3.}$ ${\displaystyle 2}$ ${\displaystyle 0}$ ${\displaystyle 3/8}$

${\displaystyle {\tfrac {3}{8}}=0.011_{2}\mapsto 0.11_{2}\mapsto 0.1_{2}\mapsto 0\mapsto 0\mapsto \cdots }$

Looking at the quantities

${\displaystyle P\,=\{x\in [0,1):x\equiv {\tfrac {a}{u}}{\text{ mit }}u\in \mathbb {N} {\text{ ungerade und }}a\in \mathbb {N} _{0}{\text{ teilerfremd zu }}u\}}$

${\displaystyle W=\{x\in [0,1):x\equiv {\tfrac {a}{2^{n}}}{\text{ mit }}a,n\in \mathbb {N} _{0}\}}$,

then you can see immediately that the set of periodic points is because the decimal places of the elements of are periodic. The set of periodic points - these are the rational numbers with an odd denominator - lie close to the interval. With the above definition, the interval corresponds to the Julia set of The Julia set of is therefore the edge of the unit circle ${\displaystyle P}$ ${\displaystyle g}$ ${\displaystyle P}$ ${\displaystyle [0,1).}$ ${\displaystyle f.}$ ${\displaystyle f}$ ${\displaystyle S_{1}:}$

${\displaystyle J_{f}=S_{1}=\{z\in {\overline {\mathbb {C} }}:|z|=1\}}$

All elements of are finally mapped to zero, because the elements of have a terminating dual development. so is the inverse orbit of below . According to property 5, this set is dense in the Julia set: The numbers with a terminating dual expansion are close in the interval The Julia set is both the edge of the catchment area of and the edge of the catchment area of (property 6). ${\displaystyle W}$ ${\displaystyle W}$ ${\displaystyle W}$ ${\displaystyle 0}$ ${\displaystyle g}$ ${\displaystyle [0,1).}$ ${\displaystyle \infty }$ ${\displaystyle 0}$

Property 7 can also be demonstrated directly: Be a neighborhood of a point of, i.e. a part of the unit circle of length If the length is smaller than the semicircle, then the length of the part doubles with each application of One choose so that applies and has covers the entire Julia set. ${\displaystyle U}$ ${\displaystyle J_{f},}$ ${\displaystyle \ell .}$ ${\displaystyle f.}$ ${\displaystyle n}$ ${\displaystyle \ell \cdot 2^{n}\geqslant 2\pi }$

All rational numbers lead to sequences that eventually become periodic. The reason for this is that rational numbers have a periodic dual expansion. Correspondingly, irrational numbers lead to sequences that do not become periodic.

Dynamics of quadratic polynomials

The Julia set for The Transformation can be extended to the whole purple part of the Fatou set. ${\displaystyle z\mapsto z^{2}-1:}$ ${\displaystyle \varphi _{c}}$

The Julia set for consists of Cantor dust. cannot be extended to the whole purple area. ${\displaystyle z\mapsto z^{2}-0{,}6+0{,}6\mathrm {i} :}$ ${\displaystyle \varphi _{c}}$

In the general case of quadratic polynomials, it is sufficient to use polynomials of the shape

${\displaystyle f_{c}\colon z\mapsto z^{2}+c}$

to be considered, because all other quadratic polynomials can be brought into this representation by a linear coordinate transformation.

As with the normal parabola, there is an attractive fixed point in the figure, and in a neighborhood of there is a transformation that converts into a normal parabola: ${\displaystyle \infty }$ ${\displaystyle \infty }$ ${\displaystyle \varphi _{c},}$ ${\displaystyle f_{c}}$

${\displaystyle \varphi _{c}^{-1}\circ f_{c}\circ \varphi _{c}=z^{2}}$

If a point lies in this environment and is reversible there, the archetype can be found for the point using the iteration rule : ${\displaystyle z_{n}}$ ${\displaystyle f_{c}}$ ${\displaystyle z_{n-1}}$

${\displaystyle z_{n-1}:={\sqrt {z_{n}-c}}}$

The archetype is selected in such a way that the transformation can be continued steadily to the new, larger area. With this method, the environment in which the same dynamic as has can be gradually enlarged - at least as long as the function can be reversed, as long as one does not reach a critical point of the function through backward iteration. The behavior of the critical point is therefore decisive for the dynamics. This is the only critical point except ${\displaystyle \varphi _{c}}$ ${\displaystyle f_{c}}$ ${\displaystyle z^{2},}$ ${\displaystyle 0.}$ ${\displaystyle \infty .}$

If it is in the catchment area of then the transformation can at some point no longer be continued because the backward iteration arrives at this point of the non-reversibility of . If the point does not strive against it , then the homeomorphism can be extended to all points outside the circular disk. In this case the Julia set of is connected. ${\displaystyle 0}$ ${\displaystyle \infty ,}$ ${\displaystyle f_{c}}$ ${\displaystyle 0}$ ${\displaystyle \infty }$ ${\displaystyle \varphi _{c}}$ ${\displaystyle f_{c}}$

If, on the other hand, is in the catchment area of then the transformation cannot be extended to the circular disk because one arrives at a branching point, namely the critical point. In this case there can be no other attractive attractor besides the attractor , because every attractive attractor contains at least one critical point. In this case the Julia set consists of Cantor dust and the Fatou set has only one connected component . ${\displaystyle 0}$ ${\displaystyle \infty ,}$ ${\displaystyle \infty }$

For the Lebesgue measure of the Julia set of rational mappings, it was assumed for a long time, according to the examples in which it could be calculated, that it either is (Cantor-Staub) or encompasses the entire Riemann sphere. The existence of Julia sets of positive Lebesgue measures in the iteration of quadratic polynomials was conjectured by Adrien Douady and proved in 2005 by Xavier Buff and Arnaud Chéritat . ${\displaystyle 0}$

Relationship to the Mandelbrot crowd

Julia sets for different parameters reveal the Mandelbrot set.

These two fundamentally different properties give rise to the definition of a parameter set that contains all complex numbers for which the critical point from not to escapes: the Mandelbrot set ${\displaystyle 0}$ ${\displaystyle f_{c}}$ ${\displaystyle \infty }$

${\displaystyle M:=\left\{c\in {\overline {\mathbb {C} }}:f_{c}^{n}(0)\not \to \infty {\text{, wenn }}n\to \infty \right\}}$

That is, the Mandelbrot set is the set of parameters for which the recursion remains constrained if one chooses. ${\displaystyle c,}$ ${\displaystyle z_{n+1}=z_{n}^{2}+c}$ ${\displaystyle z_{0}=0}$

The Mandelbrot set is therefore a description of the Julia sets of quadratic polynomials. A Julia set corresponds to each point of the complex number plane. Properties of the Julia set can be judged by the position of relative to the Mandelbrot set: If the point is an element of the Mandelbrot set, then both the Julia set and are connected. Otherwise both Cantor sets are disconnected points. If the point is then the Fatou set consists of two related components, namely the area bounded by the Julia set and the catchment area of Is not in the Mandelbrot set, then the Fatou set only consists of the catchment area of ${\displaystyle c}$ ${\displaystyle c}$ ${\displaystyle c}$ ${\displaystyle J_{c}}$ ${\displaystyle K_{c}}$ ${\displaystyle M,}$ ${\displaystyle \infty .}$ ${\displaystyle c}$ ${\displaystyle \infty .}$

If near the edge of the Mandelbrot set, then the corresponding Julia set resembles the structures of the Mandelbrot set in the vicinity of ${\displaystyle c}$ ${\displaystyle c.}$

Graphic representation of the Julia sets

For plotting the filled Julia sets in the two-dimensional complex plane, the color of a point is selected according to how many iterations were needed before because the iteration for all with divergent. Points whose absolute value is smaller than after a specified maximum number of iteration steps are assumed to be converging and are usually shown in black. The choice of is possible, however, arise for larger values as harmonious coloring, which also good to equipotentials an electrically charged corresponding Julia set. ${\displaystyle K_{c}}$ ${\displaystyle |z_{n}|\geq K\geq 2,}$ ${\displaystyle z}$ ${\displaystyle |z|\geq 2}$ ${\displaystyle K}$ ${\displaystyle K=2}$ ${\displaystyle K=1000}$

The general definition

The above method cannot be used for holomorphic or meromorphic functions that are not polynomials, since the iterated function values ​​generally do not approach infinity for a single initial value. There are several ways to define the Julia set for such general functions: ${\displaystyle f,}$ ${\displaystyle J(f)}$

• ${\displaystyle J(f)}$is then the smallest infinite and closed subset of the complex plane that is invariant under , i.e. that is, whose image and archetype are again entirely contained in the set. For example, for every polynomial of degree over the complex numbers, the boundary of the set is closed, infinitely large and invariant under. Therefore it must contain the Julia set of . That the edge is indeed equal to the Julia set still requires some work. ${\displaystyle f}$ ${\displaystyle P(z)}$ ${\displaystyle \geqslant 2}$ ${\displaystyle \{z\mid {\text{Die Folge }}(p^{n}(z))_{\in \mathbb {N} }{\text{ ist beschränkt.}}\}}$ ${\displaystyle p(z).}$ ${\displaystyle p(z)}$
• ${\displaystyle J(f)}$is the set of points for which the family of iterated functions is not uniformly continuous on every compact subset of . Concretely: If there is a given one so that in every environment, no matter how small, there is a point for which the iterated values and at some point have a distance greater than , then one of the Julia set belongs to Here, however, the complex number level cannot be mixed with the Euclidean Metric provided, rather the complex numbers have to be understood as a Riemann number sphere and provided with the corresponding spherical metric. According to Arzelà-Ascoli's theorem , the latter definition is equivalent to Fatous’s definition of the Julia set: Let be a rational (or meromorphic) function on the Riemann number sphere Then a point is called a normal point of if the family of iterates in an open neighborhood of the point is one normal family (in the sense of Montel ) forms. We call the set of all normal points the Fatou set and its complement the Julia set of ${\displaystyle f^{n}}$ ${\displaystyle J(f)}$ ${\displaystyle x_{0}\in \mathbb {C} }$ ${\displaystyle \epsilon >0,}$ ${\displaystyle x_{0}}$ ${\displaystyle z}$ ${\displaystyle f^{n}(z_{0})}$ ${\displaystyle f^{n}(z)}$ ${\displaystyle \epsilon }$ ${\displaystyle z_{0}}$ ${\displaystyle f.}$ ${\displaystyle f}$ ${\displaystyle \mathbb {C} \cup \{\infty \}.}$ ${\displaystyle z_{0}\in \mathbb {C} \cup \{\infty \}}$ ${\displaystyle f,}$ ${\displaystyle \{f^{n}\mid n\in \mathbb {N} \}}$ ${\displaystyle z_{0}}$ ${\displaystyle F(f)}$ ${\displaystyle (\mathbb {C} \cup \{\infty \})\setminus F(f)}$ ${\displaystyle J(f)}$ ${\displaystyle f.}$

generalization

One can also extend the original definition to the algebra of quaternions . This is a real four-dimensional space , which is why a complete representation of a Julia set in it is problematic. But it is possible to visualize the intersection of such a Julia set with a three-dimensional hyperplane.

Example Pictures

• 

  ? Animated zoom

• 

  ? Animation over the parameter ${\displaystyle c}$

• 

  Julia set for a third degree polynomial

• 

  Julia set in the space of quaternions

See also

• Newton fractal

literature

• Alan F. Beardon: Iteration of rational functions . Springer, 1991.
• Norbert Steinmetz: Rational iteration . Walter de Gruyter, 1993.
• John Milnor : Dynamics in one complex variable . Princeton University Press, 2006, arxiv : math.DS / 9201272 .
• Christoph Dötsch: Dynamics of meromorphic functions on the Riemann number sphere. To characterize Julia sets . Diplomica Verlag , 2008, ISBN 3-8366-6026-1 ( Google Books ).

Web links

Wiktionary: Julia set  - explanations of meanings, word origins, synonyms, translations

Commons : Julia crowd  - album with pictures, videos and audio files

• Explore Julia fractals (with Java applets )

Individual evidence

1. ^ A. Cayley: The Newton-Fourier imaginary problem . Amer J Math II 97, 1879.
2. ^ P. Blanchard: Complex Analytic Dynamics on the Riemann Sphere . Bull Amer Math Soc 11th , online. ( Memento of the original from October 28, 2017 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.