Julia set

The Julia sets, first described by Maurice Gaston Julia and Pierre Fatou, are subsets of the complex plane, where at any holomorphic or meromorphic function a Julia set belongs. Often the Julia sets are fractal sets. The complement of the Julia set is called the Fatou set.

Applying a defined function on all over again on their function values ​​, is then obtained for each of a sequence of complex numbers:

Depending on the initial value, this sequence show two fundamentally different behavior:

• A small change in the initial value leads to practically the same result, the dynamics is stable in a certain sense, the starting value is assigned to the Fatou set.
• A little change of the starting value leads to a completely different behavior of the sequence, the dynamics depends "chaotic" from the start value: the initial value belongs to the Julia set.

• 3.1 Dynamics on the example z → z2
• 3.2 Dynamics of quadratic polynomials
• 3.3 Relationship to the Mandelbrot set
• 3.4 Graphical representation of the Julia sets

Background

Newton's method is one of the best known and most widely used methods for solving nonlinear equations. If one has the equation to be solved in the form

Written, then the zeros z are p to find a function. If the function p is differentiable, then Newton's method transforms the static problem " p ( x ) = 0" in a dynamic process: it provides an iteration of the form

With the following properties

• The zeros of p are to attracting fixed points of f
• If the start value of the iteration z0 close to a root of p, then the Newton iteration converges to the fixed point of f belonging, and thus against this zero point.

All you need is an approximate solution of the problem then. The fixed points then act like the centers of force fields that attract particles in their vicinity. With each iteration, the particles move closer to the power source.

As other fixed point iterations also - - From its conception it is the Newton method so a local process whose behavior is known, if one is close to a zero. But what happens if we continue to move away from the attractions, and what are the boundaries between the catchment areas of individual power sources?

Serious studies on the global dynamics of the process goes back to the year 1879, when Lord Arthur Cayley extended the problem of the real numbers to the complex numbers and global studies suggested:

He, however, already encountered for the case where p is a polynomial of third degree, insurmountable problems, so he finally stopped its investigation:

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

With a meromorphic function f given but not the standard metric of the level will be used. Instead, the complex plane is projected by means of a stereographic projection of the surface of a sphere. Each point on the sphere besides the pole corresponds to a complex number. By including the North Pole, which is identified with ∞, arises the Riemann sphere. The metric is then the distance between two points on the spherical surface.

Properties

Let f be a meromorphic function that is on the completion of complex numbers, ie the quotient of two relatively prime polynomials. In addition, the degree of f is greater than one. The degree of a meromorphic function is the maximum of the degrees of coprime polynomials in the numerator and denominator. The degree indicates in general how many archetypes has a point. Depending shows the dynamics of the process for a given initial value, that value is assigned to one of two amounts:

The Riemann sphere is the disjoint union of these two quantities. Thus, each point belongs to either the Fatou set or the Julia set. The Julia set of a function is referred to as and the Fatou set as.

The historical definition of the Julia set, as it comes from Fatou and Julia, and can be read below is not particularly intuitive or clear. Therefore, here are some properties of these sets are assembled, including first a few basic terms are used.

Terms

For each value defines the recursion

A sequence of points on the Riemann sphere. This sequence is referred to as orbit:

It always means n times sequential execution of f and is not to be confused with the n- th power. The definition of the inverse orbit done something different because f is not uniquely invertible in general. The inverse orbit of a point consists of all points that are eventually mapped to this:

If for some n holds, then is called a periodic point and the orbit

Is called periodic orbit or cycle. If n is the smallest natural number with this property, then n is called the period of the cycle. If this is the case for, so if applies then a fixed point of f is obviously a periodic point of f, the period of which is equal to n, of a fixed point. Based on the derivation can characterize the stability of a periodic point. to this end let

Then called the periodic point

• Strongly attractive when
• Attractive when
• Indifferent if
• Repulsive when

By applying the chain rule, we see that () ' for all points of the cycle has the same value, and similarly it means this cycle (strong) attracting, indifferent or repelling.

This naming is motivated by the following observation: In the case of the fixed point behaves in an environment just like in a neighborhood of zero. Under iteration values ​​therefore always moving closer to the fixed point approach, if valid, and for the values ​​further and further away from the fixed point. Under the iteration of the fixed point, thus encompassing values ​​in his environment or he rejects them. For the case is more complicated, and the values ​​are at least as strong as attracted by in a neighborhood of 0

Is an attractive fixed point of f, then that means the amount

The catchment area of the fixed point. The amount thus consists of all points whose orbit converges to. Apparently, this set contains the inverse orbit. The A comes from the English basin of attraction ( catchment area / collection basin of the attractor, so here reservoir an attracting fixed point or cycle). If an attracting periodic cycle of period, then each of the fixed points has its catchment area, and refers to the union of these basins.

Definition

One possible definition of the Julia set is done by the amount of their repelling periodic points:

With completion means the topological degree. This is the definition, on the Julia built his theory. The starting point of Fatouschen work was another specified below definition.

Each element of the Julia set so can be represented as the limit of a convergent sequence, which consists only of repelling periodic points of f.

Basic Properties

Some properties of the Julia set are:

Notes

Critical points

A point is called a critical point of if in any neighborhood is reversible. Is differentiable, then a critical point by

Characterized. In each catchment, which belongs to a (highly ) attractive attractor is at least one critical point. By looking at the critical points of a function, therefore, statements about the dynamics of this function can be made.

A well-known example is the Mandelbrot set, which is explained with reference to certain Julia sets below. The Mandelbrot set mapped the different behavior of the critical point 0 of the mapping z → z ² c for various values ​​of c.

Julia sets of polynomials

A simple way, the Julia set of a polynomial p to define, is the following recursion:

Z0 with a start value.

The quantity Kp is defined as the set of all complex numbers z0, the amount of which is limited by any number of iterations. The Julia set Jp is then the boundary of this set. Kp is also imprecise as filled Julia set or occasionally as a Julia set itself. One can prove that Kp is limited.

This definition is the direct conversion of the property 6: For a polynomial ∞ is an attracting fixed point. The Julia set that is obtained as the edge of the catchment area of this fixed point. If a point lies in, then he finally converges to ∞ or - when using the standard metric - its magnitude grows beyond all limits. If its amount is limited, then it belongs to the basin of another attractor or the Julia set itself

This " definition " is generally used to generate graphics, because they can easily be translated into a computer program.

For meromorphic functions whose numerator is greater than its denominator by at least 2 degrees, you can use the same definition, there is an attractive fixed point for such functions ∞.

Dynamics using the example of z → z2

This simple example already can prove many properties of the Julia set.

The function has three fixed points: 0, 1 and ∞. For these points. Since the derivative vanishes at 0 and ∞, these two fixed points are attracting fixed points, while one is repulsive. All initial values ​​whose magnitude is less than 1, converge to 0, and all initial values ​​whose magnitude is greater than 1, converge to ∞:

In the remaining case lies on the unit circle and has the representation, and it is. Application of F so doubles only the (real) in the exponent of the polar coordinates representation, the magnitude of the number remains equal to 1, the exponent x can always be chosen so that it is in the semi-open interval [0,1). Considering only the effect of f on the variable x in the exponent, then f corresponds to the figure

To the interval [0,1 ) of the real numbers, that is a multiplication by 2, and only the fractions are relevant. The fixed point of f 1 is the fixed point of 0. Iterates to the value 1 /3, then the sequence results

Thus, 1/3, a periodic point, as 2/3. In the representation of a number as a binary fraction only the digits are determined by the multiplication by 2 is shifted one position to the left, and the decimal point is always set by the mod to 0, as shown in Example 3 /8:

Regarding the amounts of

Then one sees directly that P is the set of periodic points of because the decimal places of the elements of P are periodic. The set of periodic points - these are the rational numbers with odd denominators - are close in the interval. With the above definition corresponds to the interval of the Julia set of f, the Julia set of f is thus the boundary of the unit circle:

All elements of W are eventually mapped to zero, because the elements of W have a dual -terminating development. So under the inverse orbit of 0. According to property 5, this amount is dense in the Julia set: the numbers with aborting dual development are dense in the interval [0,1). The Julia set is both the edge of the basin of ∞ and the edge of the catchment area of zero ( property 6).

Property 7 can also prove directly: Let U be a neighborhood of a point of, that is a section of the unit circle of length. If the length is less than the half-circle, then the length of the section doubles with each application of f We therefore choose the way that is true, and has the entire Julia set covers.

All rational numbers lead to consequences that are ultimately periodic. Reason for this is that rational numbers have periodic Dual development. According irrational numbers lead to consequences that are not periodic.

Dynamics of quadratic polynomials

In the general case of quadratic polynomials, it suffices polynomials of the form

Consider, for all other quadratic polynomials can be brought by a linear coordinate transformation in this representation.

As with the standard parabola ∞ is an attracting fixed point of the figure, and in a neighborhood of ∞, there is a transformation which converts fc in a standard parabola:

If there is a point zn in this environment and there is fc reversible, then can be to the point by means of the iteration the prototype of zn -1, see:

The archetype is selected so that the transformation can be continued steadily to the new, larger area. By this method, the environment in which fc is the same as Z2 dynamics can be increased gradually - at least as long as the function can be reversed, ie it does not pass through a critical point of the backward iteration function. Crucial for the dynamics is therefore the behavior of the critical point 0, which is the only critical point except ∞.

Located in the catchment area of 0 ∞, then the transformation can no longer be continued sometime, because the backward iteration reaches this point, the non- reversibility of fc. If the point 0 does not tend to ∞, then the homeomorphism can be extended for all points outside the circular disk. In this case, the Julia set of fc is connected.

If 0, however, in the catchment area of ∞, then the transformation can not be extended to the circular disk, because at a branch point, namely the critical point reached. In this case, there may be next to the attractor ∞ no other attractive attractor, because every attractive attractor contains at least one critical point. In this case, the Julia set of Cantor dust and the Fatou set has only one connected component.

Relationship with the Mandelbrot set

These two fundamentally different properties give rise to the definition of a parameter set, which includes all complex numbers for which the critical point 0 of fc not escaping to ∞: the Mandelbrot set

That is, the Mandelbrot set is the set of parameter c for which the recursion zn 1 = Zn2 c remains limited when z0 = 0 selects.

The Mandelbrot set is thus a description quantity of Julia sets of quadratic polynomials. Each point c of the complex plane corresponds to a Julia set. Properties of the Julia set can be able to judge of c relative to the Mandelbrot set: If the point c is an element of the Mandelbrot set, then both the Julia set Jc and Kc are contiguous. Otherwise, both Cantor sets are disjoint points. If the point in M, then there is the Fatou set of two Zusammenhangskomponeten, namely from the space defined by the Julia set area and the catchment area of ∞. If c is not in the Mandelbrot set, then there is a Fatou amount only from the catchment area of ∞.

If c is close to the edge of the Mandelbrot set, then similar to the corresponding Julia set the structures of the Mandelbrot set in the vicinity of c.

Graphical representation of the Julia sets

For graphical representation of the filled Julia set Kc in the two-dimensional complex plane, the color of a point is chosen according to how many iterations were needed before, since the iteration for all z diverges. Points that are smaller in magnitude than K after a predetermined maximum number of iteration steps are assumed to be convergent and shown in black in the rule. The choice of K = 2 is possible, however, arise for larger values ​​of K = 1000 harmonious colorations also well correspond to the equipotential lines of an electrically charged Julia set.

The general definition

For holomorphic or meromorphic functions that are not polynomials, the above method can not be applied because the iterated function values ​​in general for a single initial value tend to infinity. There are several ways to define the Julia set of such general features:

• J ( f ) is then the smallest infinite and closed subset of the complex plane, which is invariant under f, that is, whose image and inverse image is again entirely contained in the set. For example, for any polynomial p (z ) of degree ≥ 2 over the complex numbers of the boundary of the set { z | } The sequence is bounded completed, infinitely large and invariant p (z). That's why he needs the Julia set of p ( z) included. The fact that the edge is in fact equal to the Julia set, but still requires some work.
• Y ( f) is the set of points, in which the family of the iterated function ( f) n is not equicontinuous each compact subset of Y ( f). Specifically: Are there too given, so that in every small area around x0 is a point z is, for the fn the iterated values ​​(z0 ) and fn ( z) at some point a distance have large, so z0 belongs to the Julia set of f Here, one may, however, the complex plane not equipped with the Euclidean metric, but you have to understand the complex numbers as a Riemann sphere and with the corresponding spherical metric.

By the theorem of Arzela - Ascoli latter definition is equivalent to the Fatouschen definition of the Julia set: Let f be a rational (or meromorphic ) function on the Riemann sphere. Then is called a point normal point of f, if the family of iterates in an open neighborhood of the point a normal family forms (in the sense of Montel ). The set of all regular points we call the Fatou set F ( f), and its complement we call the Julia set J ( f) of f

Generalization

One can also extend the original definition of the algebra of quaternions. This is a real four-dimensional space to undergo a full representation of a Julia set is problematic in it. But it is possible to visualize the section of such a Julia Set with a three-dimensional hyperplane.

Imaging

Animation via the parameter c

Julia set for a third degree polynomial