Lots of Julia

A lot of Julia. More precisely, this is not the set itself (which in this case consists of disconnected points and cannot be drawn), but points from its vicinity. The brighter the point, the closer it is to the Julia set and the more iterations it needs to go from zero to a given large distance

A lot of Julia. More precisely, this is not the set itself (which in this case consists of disconnected points and cannot be drawn), but points from its vicinity. The brighter the point, the closer it is to the Julia set and the more iterations it needs to go from zero to a given large distance

The filled Julia set for the map f ( z ) = z ² −1. Axial symmetry indicates the absence of an imaginary component in the free term of the mapping f ( z )

The filled Julia set for the map f ( z ) = z ² + 0.28 + 0.0113 i . Anticlockwise swirls indicate a positive imaginary component in the free term of the map f ( z )

In holomorphic dynamics , the Julia set $J(f)$ ${\ displaystyle J (f)}$ $J (f)$ rational display $f:\mathbb {C} P^{1}\to \mathbb {C} P^{1}$ ${\ displaystyle f: \ mathbb {C} P ^ {1} \ to \ mathbb {C} P ^ {1}}$ ${\ displaystyle f: \ mathbb {C} P ^ {1} \ to \ mathbb {C} P ^ {1}}$ - many points, the dynamics in the vicinity of which are in a certain sense unstable with respect to small perturbations of the initial position. If f is a polynomial, we also consider the filled Julia set - the set of points that do not tend to infinity. The usual Julia set is at the same time its boundary .

Many Fatou $F(f)$ ${\ displaystyle F (f)}$ $F (f)$ - addition to the Julia set. In other words, the dynamics of iteration f on $F(f)$ ${\ displaystyle F (f)}$ $F (f)$ regular but on $J(f)$ ${\ displaystyle J (f)}$ $J (f)$ chaotic.

Supplements Picard ’s large theorem on “the behavior of an analytic function in a neighborhood of an essentially singular point”.

These sets are named after the French mathematicians Gaston Julia and Pierre Fatou , who laid the foundation for the study of holomorphic dynamics at the beginning of the 20th century.

Content

Definitions

Let be $f:\mathbb {C} P^{1}\to \mathbb {C} P^{1}$ ${\ displaystyle f: \ mathbb {C} P ^ {1} \ to \ mathbb {C} P ^ {1}}$ - rational mapping. The Fatou set consists of points z such that, in the restriction on a sufficiently small neighborhood z, the sequence of iterations

(f^{n})_{n\in \mathbb {N} }

{\ displaystyle (f ^ {n}) _ {n \ in \ mathbb {N}}}

forms a normal family in the sense of Montel . The Julia set is an addition to the Fatou set.

This definition allows the following equivalent reformulation: the Fatou set is the set of those points whose orbits are Lyapunov stable . (The equivalence of the reformulation is not obvious, but it follows from Montel’s theorem .)

Properties

As follows from the definitions, the Julia set is always closed , and the Fatou set is open .
The Julia set to display a degree greater than 1 is always nonempty (otherwise it would be possible to choose a uniformly convergent subsequence from iterations.) For the Fatou set, a similar statement is false: there are examples in which the Julia set turns out to be the entire Riemann sphere . Such an example can be constructed by taking a mapping $z\mapsto 2z(mod\,\mathbb {Z} [i])$ ${\ displaystyle z \ mapsto 2z (mod \, \ mathbb {Z} [i])}$ doubling on the torus $\mathbb {C} /\mathbb {Z} [i]$ ${\ displaystyle \ mathbb {C} / \ mathbb {Z} [i]}$ (whose dynamics is obviously random everywhere) and passing it through $\wp$ ${\ displaystyle \ wp}$ -weierstrass function $\wp :\mathbb {C} /\mathbb {Z} [i]\to \mathbb {C} P^{1}$ ${\ displaystyle \ wp: \ mathbb {C} / \ mathbb {Z} [i] \ to \ mathbb {C} P ^ {1}}$ .
The Julia set is the closure of the union of all repulsive periodic orbits.
The Fatou and Julia sets are both completely invariant under the action of f , that is, they coincide both with their image and with the full inverse image:

\ f^{-1}(J(f))=f(J(f))=J(f),

{\ displaystyle \ f ^ {- 1} (J (f)) = f (J (f)) = J (f),}

\ f^{-1}(F(f))=f(F(f))=F(f).

{\ displaystyle \ f ^ {- 1} (F (f)) = f (F (f)) = F (f).}

The Julia set J (F) is the boundary of the (complete) basin of attraction of any attracting or super-attracting orbit; A special case of this is the statement that J (F) is the boundary of the filled Julia set (since for a polynomial map, infinity is a super-attracting fixed point, and the filled Julia set is an addition to its attraction basin). In addition, taking a polynomial map with three different attracting fixed points, we obtain an example of three open (naturally, disconnected) sets on a plane with a common boundary.
If an open set $U$ ${\ displaystyle U}$ intersects the Julia set, then, starting from some sufficiently large n , the image $f^{n}(U\cap J)=f^{n}(U)\cap J$ ${\ displaystyle f ^ {n} (U \ cap J) = f ^ {n} (U) \ cap J}$ matches the whole set of julia $J$ ${\ displaystyle J}$ . In other words, iterations stretch an arbitrarily small neighborhood in the Julia set over the entire Julia set.
Since the aforementioned stretching most often occurs rather quickly, holomorphic mappings are conformal , and the Julia set is invariant with respect to dynamics — it turns out to have a fractal structure : its small parts resemble large ones.
If the Julia set is different from the entire Riemann sphere , then it has no interior points .
For all points z of the Riemann sphere, except maybe two, the set of limit points of the sequence of complete inverse images $f^{-n}(z)$ ${\ displaystyle f ^ {- n} (z)}$ there are many Julia. This property is used in computer algorithms for constructing the Julia set.
Sullivan's theorem states that any connected component of a Fatou set is pre-periodic. In turn, the classification theorem for the periodic components of a Fatou set claims that periodic components are of one of four types: an attraction pool of an attracting or super-attracting fixed or periodic point, a Fatou lobe of a parabolic point, a Siegel disk, and an Erman ring .

Related Concepts

Quadratic mapping $z\mapsto P_{2}(z)$ ${\ displaystyle z \ mapsto P_ {2} (z)}$ changing coordinates always reduces to $z\mapsto z^{2}+c$ ${\ displaystyle z \ mapsto z ^ {2} + c}$ . It turns out that the Julia set will be connected if and only if the critical point z = 0 (or, equivalently, its image z = c ) does not go to infinity. If iterations 0 tend to infinity, the Julia set (which coincides, in this case, with the filled Julia set) turns out to be homeomorphic to the Cantor set and has measure zero. In this case, it is called the Fatou dust (despite the confusing name, this is precisely the Julia set - a lot of chaotic dynamics!).

The set of parameters c for which the Julia set of quadratic dynamics is connected is called the Mandelbrot set . It also has a fractal structure (and is probably one of the most famous fractals).

Numerical construction

Border Scan Method (BSM)

If the function f has several attractors (fixed or periodic attracting points), the Julia set is the boundary of the attraction basin of any of them. This property is based on the algorithm for constructing the image of the Julia set, called the "boundary scanning method" (BSM). It consists of the following. Consider a grid of rectangular pixels. To determine whether to paint a pixel as belonging to the Julia set, the image of each of its “angles” is calculated under the influence of a large number of iterations f. If the images are far from each other, then the angles belong to the pools of different attractors. It follows that the border between the pools passes through this pixel, and it is painted over. Going through all the pixels, we get an image approximating the Julia set.

This method can also be used when there are no two attractors, but there are Siegel disks , Erman rings, or parabolic pools. (If two close points remain close, then their orbits are Lyapunov stable, and a small neighborhood of these points belongs to the Fatou region; otherwise, there are points of the Julia set near them.) At the same time, this method does not work when the map has only one attractor , and almost the entire sphere of Riemann is its basin of attraction. (For example, $z\mapsto z^{2}+i$ ${\ displaystyle z \ mapsto z ^ {2} + i}$ .) ^[1]

Inverse Iteration Method (IIM)

The Julia set is the closure of the union of all complete inverse images of any repelling fixed point. Thus, if there is an effective inverse mapping algorithm $f^{-1}$ ${\ displaystyle f ^ {- 1}}$ , and at least one repulsive fixed point is known, in order to construct the Julia set, its inverse images can be sequentially calculated. At each step, each point has as many prototypes as the degree of f, so the total number of prototypes grows exponentially, and storing their coordinates requires large amounts of memory. ^[1] In practice, the following modification is also used: at each step, one random prototype is selected. At the same time, however, it must be borne in mind that such an algorithm bypasses the Julia set not uniformly: it can get into some areas only in a very large (almost unattainable) time, and they will not be shown on the resulting graph.

Interesting Facts

Mathematicians have proved that an arbitrary closed figure on a plane can be arbitrarily close approximated by the Julia set for a suitable polynomial. Among other things, as a demonstration of their own technology, scientists managed to build a fairly good approximation of the silhouette of a cat. According to scientists, their example clearly demonstrates that the dynamics of polynomial (that is, defined by polynomials) dynamical systems can be arranged as varied as possible. They say that their proposed example will be useful in the theory of such systems ^[2]

Links

Milnor, J. Holomorphic dynamics. Introductory lectures. = Dynamics in One Complex Variable. Introductory Lectures. - Izhevsk: Research Center "Regular and chaotic dynamics", 2000. - 320 p. - ISBN 5-93972-006-4 .
A simple program for generating Julia sets (Windows, 370 kB)
Mandelbrot and Julia sets at FractalWorld

Notes

↑ ¹ ² D. Saupe. Efficient computation of Julia sets and their fractal dimension // Physica. - Amsterdam, 1987. - Vol. 28D . - S. 358-370 . Archived June 11, 2007.
↑ Mathematicians brought the cat closer by Julia sets

[Saupe-1] ¹ ² D. Saupe. Efficient computation of Julia sets and their fractal dimension // Physica. - Amsterdam, 1987. - Vol. 28D . - S. 358-370 . Archived June 11, 2007.

[2] Mathematicians brought the cat closer by Julia sets