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Julia setFrom Wikipedia, the free e
Julia set
From Wikipedia, the free encyclopedia
A Julia set
File:Julia set 3d slice animation.ogg
Three-dimensional slices through the (four-dimensional) Julia set of a function on the quaternions.
In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is 'regular', while on the Julia set its behavior is 'chaotic'.
The Julia set of a function f is commonly denoted J(f), and the Fatou set is denoted F(f).[1] These sets are named after the French mathematicians Gaston Julia[2] and Pierre Fatou[3] whose work began the study of complex dynamics during the early 20th century.
Contents [hide]
1 Formal definition
2 Equivalent descriptions of the Julia set
3 Properties of the Julia set and Fatou set
4 Examples
5 Quadratic polynomials
6 Examples of Julia sets
7 Generalizations
8 The potential function and the real iteration number
9 Field lines
10 Distance estimation
11 Plotting the Julia set
11.1 Using backwards (inverse) iteration (IIM)
11.2 Using DEM/J
12 See also
13 References
14 External links
Formal definition[edit]
Let f(z) be a complex rational function from the plane into itself, that is, f(z) = p(z)/q(z), where p(z) and q(z) are complex polynomials. Then there are a finite number of open sets F1, ..., Fr, that are left invariant by f(z) and are such that:
the union of the Fi's is dense in the plane and
f(z) behaves in a regular and equal way on each of the sets Fi.
The last statement means that the termini of the sequences of iterations generated by the points of Fi are either precisely the same set, which is then a finite cycle, or they are finite cycles of circular or annular shaped sets that are lying concentrically. In the first case the cycle is attracting, in the second it is neutral.
These sets Fi are the Fatou domains of f(z), and their union is the Fatou set F(f) of f(z). Each of the Fatou domains contains at least one critical point of f(z), that is, a (finite) point z satisfying f'(z) = 0, or z = ∞, if the degree of the numerator p(z) is at least two larger than the degree of the denominator q(z), or if f(z) = 1/g(z) + c for some c and a rational function g(z) satisfying this condition.
The complement of F(f) is the Julia set J(f) of f(z). J(f) is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers). Like F(f), J(f) is left invariant by f(z), and on this set the iteration is repelling, meaning that |f(z) - f(w)| > |z - w| for all w in a neighbourhood of z (within J(f)). This means that f(z) behaves chaotically on the Julia set. Although there are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and they make up an infinitely small part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos.
There has been extensive research on the Fatou set and Julia set of iterated rational functions, known as rational maps. For example, it is known that the Fatou set of a rational map has either 0,1,2 or infinitely many components.[4] Each component of the Fatou set of a rational map can be classified into one of four different classes.[5]
Equivalent descriptions of the Julia set[edit]
J(f) is the smallest closed set containing at least three points which is completely invariant under f.
J(f) is the closure of the set of repelling periodic points.
For all but at most two points z ∈ X, the Julia set is the set of limit points of the full backwards orbit igcup_n f^{-n}(z). (This suggests a simple algorithm for plotting Julia sets, see below.)
If f is an entire function, then J(f) is the boundary of the set of points which converge to infinity under iteration.
If f is a polynomial, then J(f) is the boundary of the filled Julia set; that is, those points whose orbits under iterations of f remain bounded.
Properties of the Julia set and Fatou set[edit]
The Julia set and the Fatou set of f are both completely invariant under iterations of the holomorphic function f:[6]
f^{-1}(J(f)) = f(J(f)) = J(f)
f^{-1}(F(f)) = f(F(f)) = F(f)
Examples[edit]
For f(z) = z^{2} the Julia set is the unit circle and on this the iteration is given by doubling of angles (an operation that is chaotic on the points whose argument is not a rational fraction of 2pi). There are two Fatou domains: the interior and the exterior of the circle, with iteration towards 0 and ∞, respectively.
For f(z) = z^{2} - 2 the Julia set is the line segment between −2 and 2. There is one Fatou domain: the points not on the line segment iterate towards ∞. (Apart from a shift and scaling of the domain, this iteration is equivalent to x o 4(x - frac{1}{2})^{2} on the unit interval, which is commonly used as an example of chaotic system.)
These two functions are of the form z^2 + c, where c is a complex number. For such an iteration the Julia set is not in general a simple curve, but is a fractal, and for some values of c it can take surprising shapes. See the pictures below.
Julia set (in white) for the rational function associated to Newton's method for f : z→z3−1. Coloring of Fatou set according to attractor (the roots of f)
For some functions f(z) we can say beforehand that the Julia set is a fractal and not a simple curve. This is because of the following result on the iterations of a rational function:
Theorem. Each of the Fatou domains has the same boundary, which consequently is the Julia set.
This means that each point of the Julia set is a point of accumulation for each of the Fatou domains. Therefore, if there are more than two Fatou domains, each point of the Julia set must have points of more than two different open sets infinitely close, and this means that the Julia set cannot be a simple curve. This phenomenon happens, for instance, when f(z) is the Newton iteration for solving the equation zn = 1 for n > 2:
f(z) = z - frac{f(z)}{f'(z)} = frac{1 + (n-1)z^n}{nz^{n-1}}.
The image on the right shows the case n = 3.
Quadratic polynomials[edit]
A very popular complex dynamical system is given by the family of quadratic polynomials, a special case of rational maps. The quadratic polynomials can be expressed as
f_c(z) = z^2 + c
where c is a complex parameter.
Filled Julia set for fc, c=1−φ where φ is the golden ratio
Julia set for fc, c=(φ−2)+(φ−1)i =-0.4+0.6i
Julia set for fc, c=0.285+0i
Julia set for fc, c=0.285+0.01i
Julia set for fc, c=0.45+0.1428i
Julia set for fc, c=-0.70176-0.3842i
Julia set for fc, c=-0.835-0.2321i
Julia set for fc, c=-0.8+0.156i
A Julia set plot showing Julia sets for different values of c; it resembles the Mandelbrot set
The parameter plane of quadratic polynomials - that is, the plane of possible c-values - gives rise to the famous Mandelbrot set. Indeed, the Mandelbrot set is defined as the set of all c such that J(f_c) is connected. For parameters outside the Mandelbrot set, the Julia set is a Cantor space: in this case it is sometimes referred to as Fatou dust.
In many cases, the Julia set of c looks like the Mandelbrot set in sufficiently small neighborhoods of c. This is true, in particular, for so-called 'Misiurewicz' parameters, i.e. parameters c for which the critical point is pre-periodic. For instance:
At c = i, the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt.
At c = −2, the tip of the long spiky tail, the Julia set is a straight line segment.
In other words the Julia sets J(f_c) are locally similar around Misiurewicz points.[7]
Examples of Julia sets[edit]
f(z) = z2 + 0.279
f(z) = z3 + 0.400
f(z) = z4 + 0.484
f(z) = z5 + 0.544
f(z) = z6 + 0.590
f(z) = z7 + 0.626
f(z) = exp(z) - 0.65
f(z) = exp(z3) - 0.59
f(z) = exp(z3) - 0.621
f(z) = z * exp(z) + 0.04
f(z) = z2 * exp(z) + 0.21
f(z) = z3 * exp(z) + 0.33
f(z) = z4 * exp(Z) + 0.41
f(z) = Sqr[Sinh(z2)] + (0.065,0.122i)
f(z) = [(z2+z)/Ln(z)] +(0.268,0.060i)
Generalizations[edit]
The definition of Julia and Fatou sets easily carries over to the case of certain maps whose image contains their domain; most notably transcendental meromorphic functions and Adam Epstein's finite-type maps.
Julia sets are also commonly defined in the study of dynamics in several complex variables.
The potential function and the real iteration number[edit]
The Julia set for f(z) = z^{2} is the unit circle, and on the outer Fatou domain, the potential function φ(z) is defined by φ(z) = log|z|. The equipotential lines for this function are concentric circles. As |f(z)| = |z|^{2} we have
varphi(z) = lim_{k oinfty} frac{log|z_k|}{2^k},
where z_k is the sequence of iteration generated by z. For the more general iteration f(z) = z^2 + c, it has been proved that if the Julia set is connected (that is, if c belongs to the (usual) Mandelbrot set), then there exist a biholomorphic map ψ between the outer Fatou domain and the outer of the unit circle such that |psi(f(z))| = |psi(z)|^{2}.[8] This means that the potential function on the outer Fatou domain defined by this correspondence is given by:
varphi(z) = lim_{k oinfty} frac{log|z_k|}{2^k}.
This formula has meaning also if the Julia set is not connected, so that we for all c can define the potential function on the Fatou domain containing ∞ by this formula. For a general rational function f(z) such that ∞ is a critical point and a fixed point, that is, such that the degree m of the numerator is at least two larger than the degree n
From Wikipedia, the free encyclopedia
A Julia set
File:Julia set 3d slice animation.ogg
Three-dimensional slices through the (four-dimensional) Julia set of a function on the quaternions.
In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is 'regular', while on the Julia set its behavior is 'chaotic'.
The Julia set of a function f is commonly denoted J(f), and the Fatou set is denoted F(f).[1] These sets are named after the French mathematicians Gaston Julia[2] and Pierre Fatou[3] whose work began the study of complex dynamics during the early 20th century.
Contents [hide]
1 Formal definition
2 Equivalent descriptions of the Julia set
3 Properties of the Julia set and Fatou set
4 Examples
5 Quadratic polynomials
6 Examples of Julia sets
7 Generalizations
8 The potential function and the real iteration number
9 Field lines
10 Distance estimation
11 Plotting the Julia set
11.1 Using backwards (inverse) iteration (IIM)
11.2 Using DEM/J
12 See also
13 References
14 External links
Formal definition[edit]
Let f(z) be a complex rational function from the plane into itself, that is, f(z) = p(z)/q(z), where p(z) and q(z) are complex polynomials. Then there are a finite number of open sets F1, ..., Fr, that are left invariant by f(z) and are such that:
the union of the Fi's is dense in the plane and
f(z) behaves in a regular and equal way on each of the sets Fi.
The last statement means that the termini of the sequences of iterations generated by the points of Fi are either precisely the same set, which is then a finite cycle, or they are finite cycles of circular or annular shaped sets that are lying concentrically. In the first case the cycle is attracting, in the second it is neutral.
These sets Fi are the Fatou domains of f(z), and their union is the Fatou set F(f) of f(z). Each of the Fatou domains contains at least one critical point of f(z), that is, a (finite) point z satisfying f'(z) = 0, or z = ∞, if the degree of the numerator p(z) is at least two larger than the degree of the denominator q(z), or if f(z) = 1/g(z) + c for some c and a rational function g(z) satisfying this condition.
The complement of F(f) is the Julia set J(f) of f(z). J(f) is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers). Like F(f), J(f) is left invariant by f(z), and on this set the iteration is repelling, meaning that |f(z) - f(w)| > |z - w| for all w in a neighbourhood of z (within J(f)). This means that f(z) behaves chaotically on the Julia set. Although there are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and they make up an infinitely small part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos.
There has been extensive research on the Fatou set and Julia set of iterated rational functions, known as rational maps. For example, it is known that the Fatou set of a rational map has either 0,1,2 or infinitely many components.[4] Each component of the Fatou set of a rational map can be classified into one of four different classes.[5]
Equivalent descriptions of the Julia set[edit]
J(f) is the smallest closed set containing at least three points which is completely invariant under f.
J(f) is the closure of the set of repelling periodic points.
For all but at most two points z ∈ X, the Julia set is the set of limit points of the full backwards orbit igcup_n f^{-n}(z). (This suggests a simple algorithm for plotting Julia sets, see below.)
If f is an entire function, then J(f) is the boundary of the set of points which converge to infinity under iteration.
If f is a polynomial, then J(f) is the boundary of the filled Julia set; that is, those points whose orbits under iterations of f remain bounded.
Properties of the Julia set and Fatou set[edit]
The Julia set and the Fatou set of f are both completely invariant under iterations of the holomorphic function f:[6]
f^{-1}(J(f)) = f(J(f)) = J(f)
f^{-1}(F(f)) = f(F(f)) = F(f)
Examples[edit]
For f(z) = z^{2} the Julia set is the unit circle and on this the iteration is given by doubling of angles (an operation that is chaotic on the points whose argument is not a rational fraction of 2pi). There are two Fatou domains: the interior and the exterior of the circle, with iteration towards 0 and ∞, respectively.
For f(z) = z^{2} - 2 the Julia set is the line segment between −2 and 2. There is one Fatou domain: the points not on the line segment iterate towards ∞. (Apart from a shift and scaling of the domain, this iteration is equivalent to x o 4(x - frac{1}{2})^{2} on the unit interval, which is commonly used as an example of chaotic system.)
These two functions are of the form z^2 + c, where c is a complex number. For such an iteration the Julia set is not in general a simple curve, but is a fractal, and for some values of c it can take surprising shapes. See the pictures below.
Julia set (in white) for the rational function associated to Newton's method for f : z→z3−1. Coloring of Fatou set according to attractor (the roots of f)
For some functions f(z) we can say beforehand that the Julia set is a fractal and not a simple curve. This is because of the following result on the iterations of a rational function:
Theorem. Each of the Fatou domains has the same boundary, which consequently is the Julia set.
This means that each point of the Julia set is a point of accumulation for each of the Fatou domains. Therefore, if there are more than two Fatou domains, each point of the Julia set must have points of more than two different open sets infinitely close, and this means that the Julia set cannot be a simple curve. This phenomenon happens, for instance, when f(z) is the Newton iteration for solving the equation zn = 1 for n > 2:
f(z) = z - frac{f(z)}{f'(z)} = frac{1 + (n-1)z^n}{nz^{n-1}}.
The image on the right shows the case n = 3.
Quadratic polynomials[edit]
A very popular complex dynamical system is given by the family of quadratic polynomials, a special case of rational maps. The quadratic polynomials can be expressed as
f_c(z) = z^2 + c
where c is a complex parameter.
Filled Julia set for fc, c=1−φ where φ is the golden ratio
Julia set for fc, c=(φ−2)+(φ−1)i =-0.4+0.6i
Julia set for fc, c=0.285+0i
Julia set for fc, c=0.285+0.01i
Julia set for fc, c=0.45+0.1428i
Julia set for fc, c=-0.70176-0.3842i
Julia set for fc, c=-0.835-0.2321i
Julia set for fc, c=-0.8+0.156i
A Julia set plot showing Julia sets for different values of c; it resembles the Mandelbrot set
The parameter plane of quadratic polynomials - that is, the plane of possible c-values - gives rise to the famous Mandelbrot set. Indeed, the Mandelbrot set is defined as the set of all c such that J(f_c) is connected. For parameters outside the Mandelbrot set, the Julia set is a Cantor space: in this case it is sometimes referred to as Fatou dust.
In many cases, the Julia set of c looks like the Mandelbrot set in sufficiently small neighborhoods of c. This is true, in particular, for so-called 'Misiurewicz' parameters, i.e. parameters c for which the critical point is pre-periodic. For instance:
At c = i, the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt.
At c = −2, the tip of the long spiky tail, the Julia set is a straight line segment.
In other words the Julia sets J(f_c) are locally similar around Misiurewicz points.[7]
Examples of Julia sets[edit]
f(z) = z2 + 0.279
f(z) = z3 + 0.400
f(z) = z4 + 0.484
f(z) = z5 + 0.544
f(z) = z6 + 0.590
f(z) = z7 + 0.626
f(z) = exp(z) - 0.65
f(z) = exp(z3) - 0.59
f(z) = exp(z3) - 0.621
f(z) = z * exp(z) + 0.04
f(z) = z2 * exp(z) + 0.21
f(z) = z3 * exp(z) + 0.33
f(z) = z4 * exp(Z) + 0.41
f(z) = Sqr[Sinh(z2)] + (0.065,0.122i)
f(z) = [(z2+z)/Ln(z)] +(0.268,0.060i)
Generalizations[edit]
The definition of Julia and Fatou sets easily carries over to the case of certain maps whose image contains their domain; most notably transcendental meromorphic functions and Adam Epstein's finite-type maps.
Julia sets are also commonly defined in the study of dynamics in several complex variables.
The potential function and the real iteration number[edit]
The Julia set for f(z) = z^{2} is the unit circle, and on the outer Fatou domain, the potential function φ(z) is defined by φ(z) = log|z|. The equipotential lines for this function are concentric circles. As |f(z)| = |z|^{2} we have
varphi(z) = lim_{k oinfty} frac{log|z_k|}{2^k},
where z_k is the sequence of iteration generated by z. For the more general iteration f(z) = z^2 + c, it has been proved that if the Julia set is connected (that is, if c belongs to the (usual) Mandelbrot set), then there exist a biholomorphic map ψ between the outer Fatou domain and the outer of the unit circle such that |psi(f(z))| = |psi(z)|^{2}.[8] This means that the potential function on the outer Fatou domain defined by this correspondence is given by:
varphi(z) = lim_{k oinfty} frac{log|z_k|}{2^k}.
This formula has meaning also if the Julia set is not connected, so that we for all c can define the potential function on the Fatou domain containing ∞ by this formula. For a general rational function f(z) such that ∞ is a critical point and a fixed point, that is, such that the degree m of the numerator is at least two larger than the degree n
0/5000
Julia setFrom Wikipedia, the free encyclopediaA Julia setFile:Julia set 3d slice animation.ogg Three-dimensional slices through the (four-dimensional) Julia set of a function on the quaternions.In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is 'regular', while on the Julia set its behavior is 'chaotic'.The Julia set of a function f is commonly denoted J(f), and the Fatou set is denoted F(f).[1] These sets are named after the French mathematicians Gaston Julia[2] and Pierre Fatou[3] whose work began the study of complex dynamics during the early 20th century.Contents [hide] 1 Formal definition2 Equivalent descriptions of the Julia set3 Properties of the Julia set and Fatou set4 Examples5 Quadratic polynomials6 Examples of Julia sets7 Generalizations8 The potential function and the real iteration number9 Field lines10 Distance estimation11 Plotting the Julia set11.1 Using backwards (inverse) iteration (IIM)11.2 Using DEM/J12 See also13 References14 External linksFormal definition[edit]Let f(z) be a complex rational function from the plane into itself, that is, f(z) = p(z)/q(z), where p(z) and q(z) are complex polynomials. Then there are a finite number of open sets F1, ..., Fr, that are left invariant by f(z) and are such that:the union of the Fi's is dense in the plane andf(z) behaves in a regular and equal way on each of the sets Fi.The last statement means that the termini of the sequences of iterations generated by the points of Fi are either precisely the same set, which is then a finite cycle, or they are finite cycles of circular or annular shaped sets that are lying concentrically. In the first case the cycle is attracting, in the second it is neutral.These sets Fi are the Fatou domains of f(z), and their union is the Fatou set F(f) of f(z). Each of the Fatou domains contains at least one critical point of f(z), that is, a (finite) point z satisfying f'(z) = 0, or z = ∞, if the degree of the numerator p(z) is at least two larger than the degree of the denominator q(z), or if f(z) = 1/g(z) + c for some c and a rational function g(z) satisfying this condition.The complement of F(f) is the Julia set J(f) of f(z). J(f) is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers). Like F(f), J(f) is left invariant by f(z), and on this set the iteration is repelling, meaning that |f(z) - f(w)| > |z - w| for all w in a neighbourhood of z (within J(f)). This means that f(z) behaves chaotically on the Julia set. Although there are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and they make up an infinitely small part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos.There has been extensive research on the Fatou set and Julia set of iterated rational functions, known as rational maps. For example, it is known that the Fatou set of a rational map has either 0,1,2 or infinitely many components.[4] Each component of the Fatou set of a rational map can be classified into one of four different classes.[5]Equivalent descriptions of the Julia set[edit]J(f) is the smallest closed set containing at least three points which is completely invariant under f.J(f) is the closure of the set of repelling periodic points.For all but at most two points z ∈ X, the Julia set is the set of limit points of the full backwards orbit igcup_n f^{-n}(z). (This suggests a simple algorithm for plotting Julia sets, see below.)If f is an entire function, then J(f) is the boundary of the set of points which converge to infinity under iteration.
If f is a polynomial, then J(f) is the boundary of the filled Julia set; that is, those points whose orbits under iterations of f remain bounded.
Properties of the Julia set and Fatou set[edit]
The Julia set and the Fatou set of f are both completely invariant under iterations of the holomorphic function f:[6]
f^{-1}(J(f)) = f(J(f)) = J(f)
f^{-1}(F(f)) = f(F(f)) = F(f)
Examples[edit]
For f(z) = z^{2} the Julia set is the unit circle and on this the iteration is given by doubling of angles (an operation that is chaotic on the points whose argument is not a rational fraction of 2pi). There are two Fatou domains: the interior and the exterior of the circle, with iteration towards 0 and ∞, respectively.
For f(z) = z^{2} - 2 the Julia set is the line segment between −2 and 2. There is one Fatou domain: the points not on the line segment iterate towards ∞. (Apart from a shift and scaling of the domain, this iteration is equivalent to x o 4(x - frac{1}{2})^{2} on the unit interval, which is commonly used as an example of chaotic system.)
These two functions are of the form z^2 + c, where c is a complex number. For such an iteration the Julia set is not in general a simple curve, but is a fractal, and for some values of c it can take surprising shapes. See the pictures below.
Julia set (in white) for the rational function associated to Newton's method for f : z→z3−1. Coloring of Fatou set according to attractor (the roots of f)
For some functions f(z) we can say beforehand that the Julia set is a fractal and not a simple curve. This is because of the following result on the iterations of a rational function:
Theorem. Each of the Fatou domains has the same boundary, which consequently is the Julia set.
This means that each point of the Julia set is a point of accumulation for each of the Fatou domains. Therefore, if there are more than two Fatou domains, each point of the Julia set must have points of more than two different open sets infinitely close, and this means that the Julia set cannot be a simple curve. This phenomenon happens, for instance, when f(z) is the Newton iteration for solving the equation zn = 1 for n > 2:
f(z) = z - frac{f(z)}{f'(z)} = frac{1 + (n-1)z^n}{nz^{n-1}}.
The image on the right shows the case n = 3.
Quadratic polynomials[edit]
A very popular complex dynamical system is given by the family of quadratic polynomials, a special case of rational maps. The quadratic polynomials can be expressed as
f_c(z) = z^2 + c
where c is a complex parameter.
Filled Julia set for fc, c=1−φ where φ is the golden ratio
Julia set for fc, c=(φ−2)+(φ−1)i =-0.4+0.6i
Julia set for fc, c=0.285+0i
Julia set for fc, c=0.285+0.01i
Julia set for fc, c=0.45+0.1428i
Julia set for fc, c=-0.70176-0.3842i
Julia set for fc, c=-0.835-0.2321i
Julia set for fc, c=-0.8+0.156i
A Julia set plot showing Julia sets for different values of c; it resembles the Mandelbrot set
The parameter plane of quadratic polynomials - that is, the plane of possible c-values - gives rise to the famous Mandelbrot set. Indeed, the Mandelbrot set is defined as the set of all c such that J(f_c) is connected. For parameters outside the Mandelbrot set, the Julia set is a Cantor space: in this case it is sometimes referred to as Fatou dust.
In many cases, the Julia set of c looks like the Mandelbrot set in sufficiently small neighborhoods of c. This is true, in particular, for so-called 'Misiurewicz' parameters, i.e. parameters c for which the critical point is pre-periodic. For instance:
At c = i, the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt.
At c = −2, the tip of the long spiky tail, the Julia set is a straight line segment.
In other words the Julia sets J(f_c) are locally similar around Misiurewicz points.[7]
Examples of Julia sets[edit]
f(z) = z2 + 0.279
f(z) = z3 + 0.400
f(z) = z4 + 0.484
f(z) = z5 + 0.544
f(z) = z6 + 0.590
f(z) = z7 + 0.626
f(z) = exp(z) - 0.65
f(z) = exp(z3) - 0.59
f(z) = exp(z3) - 0.621
f(z) = z * exp(z) + 0.04
f(z) = z2 * exp(z) + 0.21
f(z) = z3 * exp(z) + 0.33
f(z) = z4 * exp(Z) + 0.41
f(z) = Sqr[Sinh(z2)] + (0.065,0.122i)
f(z) = [(z2+z)/Ln(z)] +(0.268,0.060i)
Generalizations[edit]
The definition of Julia and Fatou sets easily carries over to the case of certain maps whose image contains their domain; most notably transcendental meromorphic functions and Adam Epstein's finite-type maps.
Julia sets are also commonly defined in the study of dynamics in several complex variables.
The potential function and the real iteration number[edit]
The Julia set for f(z) = z^{2} is the unit circle, and on the outer Fatou domain, the potential function φ(z) is defined by φ(z) = log|z|. The equipotential lines for this function are concentric circles. As |f(z)| = |z|^{2} we have
varphi(z) = lim_{k oinfty} frac{log|z_k|}{2^k},
where z_k is the sequence of iteration generated by z. For the more general iteration f(z) = z^2 + c, it has been proved that if the Julia set is connected (that is, if c belongs to the (usual) Mandelbrot set), then there exist a biholomorphic map ψ between the outer Fatou domain and the outer of the unit circle such that |psi(f(z))| = |psi(z)|^{2}.[8] This means that the potential function on the outer Fatou domain defined by this correspondence is given by:
varphi(z) = lim_{k oinfty} frac{log|z_k|}{2^k}.
This formula has meaning also if the Julia set is not connected, so that we for all c can define the potential function on the Fatou domain containing ∞ by this formula. For a general rational function f(z) such that ∞ is a critical point and a fixed point, that is, such that the degree m of the numerator is at least two larger than the degree n
If f is a polynomial, then J(f) is the boundary of the filled Julia set; that is, those points whose orbits under iterations of f remain bounded.
Properties of the Julia set and Fatou set[edit]
The Julia set and the Fatou set of f are both completely invariant under iterations of the holomorphic function f:[6]
f^{-1}(J(f)) = f(J(f)) = J(f)
f^{-1}(F(f)) = f(F(f)) = F(f)
Examples[edit]
For f(z) = z^{2} the Julia set is the unit circle and on this the iteration is given by doubling of angles (an operation that is chaotic on the points whose argument is not a rational fraction of 2pi). There are two Fatou domains: the interior and the exterior of the circle, with iteration towards 0 and ∞, respectively.
For f(z) = z^{2} - 2 the Julia set is the line segment between −2 and 2. There is one Fatou domain: the points not on the line segment iterate towards ∞. (Apart from a shift and scaling of the domain, this iteration is equivalent to x o 4(x - frac{1}{2})^{2} on the unit interval, which is commonly used as an example of chaotic system.)
These two functions are of the form z^2 + c, where c is a complex number. For such an iteration the Julia set is not in general a simple curve, but is a fractal, and for some values of c it can take surprising shapes. See the pictures below.
Julia set (in white) for the rational function associated to Newton's method for f : z→z3−1. Coloring of Fatou set according to attractor (the roots of f)
For some functions f(z) we can say beforehand that the Julia set is a fractal and not a simple curve. This is because of the following result on the iterations of a rational function:
Theorem. Each of the Fatou domains has the same boundary, which consequently is the Julia set.
This means that each point of the Julia set is a point of accumulation for each of the Fatou domains. Therefore, if there are more than two Fatou domains, each point of the Julia set must have points of more than two different open sets infinitely close, and this means that the Julia set cannot be a simple curve. This phenomenon happens, for instance, when f(z) is the Newton iteration for solving the equation zn = 1 for n > 2:
f(z) = z - frac{f(z)}{f'(z)} = frac{1 + (n-1)z^n}{nz^{n-1}}.
The image on the right shows the case n = 3.
Quadratic polynomials[edit]
A very popular complex dynamical system is given by the family of quadratic polynomials, a special case of rational maps. The quadratic polynomials can be expressed as
f_c(z) = z^2 + c
where c is a complex parameter.
Filled Julia set for fc, c=1−φ where φ is the golden ratio
Julia set for fc, c=(φ−2)+(φ−1)i =-0.4+0.6i
Julia set for fc, c=0.285+0i
Julia set for fc, c=0.285+0.01i
Julia set for fc, c=0.45+0.1428i
Julia set for fc, c=-0.70176-0.3842i
Julia set for fc, c=-0.835-0.2321i
Julia set for fc, c=-0.8+0.156i
A Julia set plot showing Julia sets for different values of c; it resembles the Mandelbrot set
The parameter plane of quadratic polynomials - that is, the plane of possible c-values - gives rise to the famous Mandelbrot set. Indeed, the Mandelbrot set is defined as the set of all c such that J(f_c) is connected. For parameters outside the Mandelbrot set, the Julia set is a Cantor space: in this case it is sometimes referred to as Fatou dust.
In many cases, the Julia set of c looks like the Mandelbrot set in sufficiently small neighborhoods of c. This is true, in particular, for so-called 'Misiurewicz' parameters, i.e. parameters c for which the critical point is pre-periodic. For instance:
At c = i, the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt.
At c = −2, the tip of the long spiky tail, the Julia set is a straight line segment.
In other words the Julia sets J(f_c) are locally similar around Misiurewicz points.[7]
Examples of Julia sets[edit]
f(z) = z2 + 0.279
f(z) = z3 + 0.400
f(z) = z4 + 0.484
f(z) = z5 + 0.544
f(z) = z6 + 0.590
f(z) = z7 + 0.626
f(z) = exp(z) - 0.65
f(z) = exp(z3) - 0.59
f(z) = exp(z3) - 0.621
f(z) = z * exp(z) + 0.04
f(z) = z2 * exp(z) + 0.21
f(z) = z3 * exp(z) + 0.33
f(z) = z4 * exp(Z) + 0.41
f(z) = Sqr[Sinh(z2)] + (0.065,0.122i)
f(z) = [(z2+z)/Ln(z)] +(0.268,0.060i)
Generalizations[edit]
The definition of Julia and Fatou sets easily carries over to the case of certain maps whose image contains their domain; most notably transcendental meromorphic functions and Adam Epstein's finite-type maps.
Julia sets are also commonly defined in the study of dynamics in several complex variables.
The potential function and the real iteration number[edit]
The Julia set for f(z) = z^{2} is the unit circle, and on the outer Fatou domain, the potential function φ(z) is defined by φ(z) = log|z|. The equipotential lines for this function are concentric circles. As |f(z)| = |z|^{2} we have
varphi(z) = lim_{k oinfty} frac{log|z_k|}{2^k},
where z_k is the sequence of iteration generated by z. For the more general iteration f(z) = z^2 + c, it has been proved that if the Julia set is connected (that is, if c belongs to the (usual) Mandelbrot set), then there exist a biholomorphic map ψ between the outer Fatou domain and the outer of the unit circle such that |psi(f(z))| = |psi(z)|^{2}.[8] This means that the potential function on the outer Fatou domain defined by this correspondence is given by:
varphi(z) = lim_{k oinfty} frac{log|z_k|}{2^k}.
This formula has meaning also if the Julia set is not connected, so that we for all c can define the potential function on the Fatou domain containing ∞ by this formula. For a general rational function f(z) such that ∞ is a critical point and a fixed point, that is, such that the degree m of the numerator is at least two larger than the degree n
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Julia set
Dari Wikipedia, ensiklopedia bebas Sebuah Julia mengatur File: Julia set 3d slice animation.ogg irisan tiga dimensi melalui (empat dimensi) Julia set fungsi pada quaternions. Dalam konteks dinamika yang kompleks, topik matematika, himpunan Julia dan set Fatou dua set melengkapi didefinisikan dari fungsi. Secara informal, yang Fatou set fungsi terdiri dari nilai-nilai dengan properti yang semua nilai terdekat berperilaku sama di bawah iterasi berulang fungsi, dan set Julia terdiri dari nilai-nilai tersebut bahwa gangguan sewenang-wenang kecil dapat menyebabkan perubahan drastis dalam urutan fungsi iterasi nilai. Dengan demikian perilaku fungsi di set Fatou adalah 'biasa', sementara di Julia mengatur perilakunya adalah 'kacau'. The Julia set fungsi f biasanya dilambangkan J (f), dan set Fatou dinotasikan F ( f). [1] set ini dinamai matematikawan Perancis Gaston Julia [2] dan Pierre Fatou [3] yang karyanya mulai mempelajari dinamika yang kompleks pada awal abad ke-20. Isi [hide] 1 Definisi Formal 2 deskripsi Setara Julia set 3 Sifat set Julia dan Fatou menetapkan 4 Contoh 5 polinomial kuadrat 6 Contoh Julia set 7 Generalisasi 8 Potensi fungsi dan iterasi bilangan real 9 baris Bidang 10 estimasi Jarak 11 Merencanakan Julia set 11.1 Menggunakan belakang (terbalik) iterasi (IIM) 11,2 Menggunakan DEM / J 12 Lihat juga 13 Referensi 14 Pranala luar Formal definisi [sunting] Biarkan f (z) adalah fungsi kompleks rasional dari pesawat ke dalam dirinya, yaitu f (z) = p (z) / q (z), dimana p (z) dan q (z) adalah polinomial kompleks. Lalu ada jumlah terbatas terbuka set F1, ..., Fr, yang tersisa invarian oleh f (z) dan sedemikian rupa sehingga: persatuan Fi adalah padat dalam pesawat dan f (z) berperilaku dalam biasa dan cara yang sama pada masing-masing set Fi. Pernyataan terakhir berarti bahwa termini dari urutan iterasi yang dihasilkan oleh titik Fi yang baik justru set yang sama, yang kemudian siklus yang terbatas, atau mereka siklus terbatas melingkar atau annular berbentuk set yang berbohong konsentris. Dalam kasus pertama siklus ini menarik, di kedua adalah netral. set ini Fi adalah domain Fatou f (z), dan serikat mereka adalah Fatou set F (f) dari f (z). Setiap domain Fatou berisi setidaknya satu titik kritis f (z), yaitu, (terbatas) titik z memuaskan f '(z) = 0, atau z = ∞, jika tingkat pembilang p (z) setidaknya dua lebih besar dari tingkat q denominator (z), atau jika f (z) = 1 / g (z) + c untuk beberapa c dan fungsi rasional g (z) memenuhi kondisi ini. Komplemen dari F (f) adalah Julia set J (f) dari f (z). J (f) adalah satu set tempat padat (itu tanpa poin interior) dan set terhitung (dari kardinalitas yang sama dengan bilangan real). Seperti F (f), J (f) yang tersisa invarian oleh f (z), dan ini mengatur iterasi yang memukul mundur, yang berarti bahwa | f (z) - f (w) |> | z - w | untuk semua w di lingkungan z (dalam J (f)). Ini berarti bahwa f (z) berperilaku berantakan di set Julia. Meskipun ada poin di Julia mengatur urutan yang iterasi terbatas, hanya ada sejumlah dihitung dari titik-titik tersebut (dan mereka membuat bagian yang sangat kecil dari set Julia). Urutan yang dihasilkan oleh titik-titik di luar himpunan ini berperilaku berantakan, fenomena yang disebut kekacauan deterministik. Telah ada penelitian yang luas pada Fatou mengatur dan Julia set fungsi rasional iterasi, yang dikenal sebagai peta rasional. Misalnya, diketahui bahwa Fatou set peta rasional memiliki baik 0,1,2 atau jauh lebih banyak komponen. [4] Setiap komponen dari Fatou set peta rasional dapat diklasifikasikan ke dalam salah satu dari empat kelas yang berbeda. [ 5] deskripsi Setara dari Julia set [sunting] J (f) adalah himpunan tertutup terkecil yang mengandung setidaknya tiga titik yang benar-benar lain di bawah f. J (f) adalah penutupan himpunan memukul mundur poin periodik. Untuk semua tapi paling banyak dua poin z ∈ X, himpunan Julia adalah himpunan titik-titik batas orbit mundur penuh igcup_n f ^? {-} n (z). (Hal ini menunjukkan algoritma sederhana untuk merencanakan Julia set, lihat di bawah.) Jika f adalah seluruh fungsi, maka J (f) adalah batas himpunan titik-titik yang menyatu hingga tak terbatas di bawah iterasi. Jika f adalah polinomial, maka J (f) adalah batas diisi Julia set; yaitu, titik-titik yang orbitnya di bawah iterasi dari f tetap dibatasi. Sifat Julia set dan Fatou set [sunting] The Julia mengatur dan Fatou set f keduanya benar-benar lain di bawah iterasi dari fungsi holomorphic f: [6] f ^ {- 1} (J (f)) = f (J (f)) = J (f) f ^ {- 1} (F (f)) = f (F (f)) = F (f) Contoh [sunting] Untuk f (z) = z ^ {2} himpunan Julia adalah lingkaran satuan dan ini iterasi diberikan dengan menggandakan sudut (sebuah operasi yang kacau pada titik-titik yang argumen bukan fraksi rasional 2pi ). Ada dua Fatou domain: interior dan eksterior lingkaran, dengan iterasi terhadap 0 dan ∞, masing-masing. Untuk f (z) = z ^ {2} - 2 set Julia adalah segmen garis antara -2 dan 2. Ada satu Fatou domain: tidak pada segmen garis poin beralih menuju ∞. (Terlepas dari pergeseran dan skala dari domain, iterasi ini setara dengan xo 4 (x -. Frac {1} {2}) ^ {2} pada interval unit, yang umumnya digunakan sebagai contoh sistem kacau) Kedua fungsi ini dari bentuk z ^ 2 + c, di mana c adalah bilangan kompleks. Untuk iterasi seperti set Julia tidak secara umum kurva sederhana, tetapi fraktal, dan untuk beberapa nilai c dapat mengambil bentuk mengejutkan. Lihat gambar di bawah ini. Julia set (putih) untuk fungsi rasional terkait dengan metode Newton untuk f: z → z3-1. Mewarnai Fatou diatur sesuai dengan penarik (akar f) Untuk beberapa fungsi f (z) kita dapat mengatakan sebelumnya bahwa set Julia adalah fraktal dan tidak kurva sederhana. Hal ini karena hasil sebagai berikut pada iterasi dari fungsi rasional: Teorema. Setiap domain Fatou memiliki batas yang sama, yang akibatnya adalah set Julia. Ini berarti bahwa setiap titik dari himpunan Julia adalah titik akumulasi untuk setiap domain Fatou. Oleh karena itu, jika ada lebih dari dua domain Fatou, setiap titik dari himpunan Julia harus memiliki poin lebih dari dua set terbuka berbeda jauh dekat, dan ini berarti bahwa himpunan Julia tidak bisa menjadi kurva sederhana. Fenomena ini terjadi, misalnya, ketika f (z) adalah iterasi Newton untuk memecahkan persamaan zn = 1 untuk n> 2: f (z) = z - frac {f (z)} {f '(z)} = frac {1 + (n-1) z ^ n} {nz ^ {n-1}}. Gambar di kanan menunjukkan kasus n = 3. polinomial kuadrat [sunting] Sebuah sistem dinamik yang sangat populer kompleks diberikan oleh keluarga polinomial kuadrat, kasus khusus dari peta rasional. Polinomial kuadrat dapat dinyatakan sebagai f_c (z) = z ^ 2 + c dimana c adalah parameter yang rumit. Diisi Julia ditetapkan untuk fc, c = 1-φ φ mana adalah rasio emas Julia ditetapkan untuk fc, c = (φ -2) + (φ-1) i = -0,4 + 0.6i Julia ditetapkan untuk fc, c = 0.285 + 0i Julia ditetapkan untuk fc, c = 0.285 + 0.01i Julia ditetapkan untuk fc, c = 0,45 + 0.1428i Julia set untuk fc, c = -0.70176-0.3842i Julia ditetapkan untuk fc, c = -0.835-0.2321i Julia ditetapkan untuk fc, c = -0,8 + 0.156i A Julia mengatur rencana yang menunjukkan Julia set untuk nilai yang berbeda dari c; menyerupai Mandelbrot set Pesawat parameter polinomial kuadrat - yaitu, pesawat kemungkinan c-nilai - menimbulkan Mandelbrot set terkenal. Memang, Mandelbrot set didefinisikan sebagai himpunan semua c sehingga J (f_c) terhubung. Untuk parameter luar Mandelbrot set, set Julia adalah ruang Cantor: dalam hal ini kadang-kadang disebut sebagai Fatou debu. Dalam banyak kasus, Julia set c terlihat seperti Mandelbrot set di lingkungan cukup kecil c. Hal ini benar, khususnya, untuk apa yang disebut 'Misiurewicz' parameter, parameter yaitu c yang titik kritis adalah pra-periodik. Misalnya: Pada c = i, yang lebih pendek, kaki depan kaki depan, set Julia tampak seperti petir bercabang. Pada c = -2, ujung ekor runcing panjang, set Julia adalah segmen garis lurus. Dengan kata lain Julia set J (f_c) secara lokal serupa di seluruh poin Misiurewicz. [7] Contoh Julia set [sunting] f (z) = z2 + 0,279 f (z) = z3 + 0.400 f (z) = z4 + 0,484 f (z) = Z5 + 0,544 f (z) = z6 + 0,590 f (z) = Z7 + 0,626 f (z) = exp (z) - 0.65 f (z) = exp (z3) - 0,59 f (z ) = exp (z3) - 0,621 f (z) = z * exp (z) + 0,04 f (z) = z2 * exp (z) + 0,21 f (z) = z3 * exp (z) + 0,33 f (z ) = z4 * exp (Z) + 0,41 f (z) = Sqr [Sinh (z2)] + (0.065,0.122i) f (z) = [(z2 + z) / Ln (z)] + (0,268, 0.060i) Generalisasi [sunting] Definisi Julia dan Fatou set dengan mudah membawa ke kasus peta tertentu yang gambarnya mengandung domain mereka; terutama transendental fungsi meromorphic dan terbatas-jenis peta Adam Epstein. Julia set juga sering didefinisikan dalam studi dinamika di beberapa variabel kompleks. Potensi fungsi dan jumlah iterasi yang nyata [sunting] The Julia ditetapkan untuk f (z) = z ^ {2} adalah lingkaran satuan, dan pada domain Fatou luar, potensi fungsi φ (z) didefinisikan oleh φ (z) = log | z |. Garis ekipotensial untuk fungsi ini adalah lingkaran konsentris. Seperti | f (z) | = | z | ^ {2} kita memiliki varphi (z) = {k lim_ oinfty} frac {log | z_k |} {2 ^ k}, di mana z_k adalah urutan iterasi yang dihasilkan oleh z . Untuk iterasi f yang lebih umum (z) = z ^ 2 + c, telah membuktikan bahwa jika himpunan Julia terhubung (yaitu, jika c milik (biasa) Mandelbrot set), maka terdapat peta ψ biholomorphic antara luar Fatou domain dan luar lingkaran satuan sehingga | psi (f (z)) | = | psi (z) |. ^ {2} [8] Hal ini berarti bahwa fungsi potensial pada domain Fatou luar didefinisikan melalui korespondensi ini diberikan oleh: . varphi (z) = {k lim_ oinfty} frac {log | z_k |} {2 ^ k} Rumus ini memiliki arti juga jika set Julia tidak terhubung, sehingga kita semua dapat c mendefinisikan fungsi potensial pada domain Fatou mengandung ∞ dengan rumus ini. Untuk fungsi rasional f umum (z) sehingga ∞ adalah titik kritis dan titik tetap, yaitu, sehingga tingkat m dari pembilang setidaknya dua lebih besar dari tingkat n
Dari Wikipedia, ensiklopedia bebas Sebuah Julia mengatur File: Julia set 3d slice animation.ogg irisan tiga dimensi melalui (empat dimensi) Julia set fungsi pada quaternions. Dalam konteks dinamika yang kompleks, topik matematika, himpunan Julia dan set Fatou dua set melengkapi didefinisikan dari fungsi. Secara informal, yang Fatou set fungsi terdiri dari nilai-nilai dengan properti yang semua nilai terdekat berperilaku sama di bawah iterasi berulang fungsi, dan set Julia terdiri dari nilai-nilai tersebut bahwa gangguan sewenang-wenang kecil dapat menyebabkan perubahan drastis dalam urutan fungsi iterasi nilai. Dengan demikian perilaku fungsi di set Fatou adalah 'biasa', sementara di Julia mengatur perilakunya adalah 'kacau'. The Julia set fungsi f biasanya dilambangkan J (f), dan set Fatou dinotasikan F ( f). [1] set ini dinamai matematikawan Perancis Gaston Julia [2] dan Pierre Fatou [3] yang karyanya mulai mempelajari dinamika yang kompleks pada awal abad ke-20. Isi [hide] 1 Definisi Formal 2 deskripsi Setara Julia set 3 Sifat set Julia dan Fatou menetapkan 4 Contoh 5 polinomial kuadrat 6 Contoh Julia set 7 Generalisasi 8 Potensi fungsi dan iterasi bilangan real 9 baris Bidang 10 estimasi Jarak 11 Merencanakan Julia set 11.1 Menggunakan belakang (terbalik) iterasi (IIM) 11,2 Menggunakan DEM / J 12 Lihat juga 13 Referensi 14 Pranala luar Formal definisi [sunting] Biarkan f (z) adalah fungsi kompleks rasional dari pesawat ke dalam dirinya, yaitu f (z) = p (z) / q (z), dimana p (z) dan q (z) adalah polinomial kompleks. Lalu ada jumlah terbatas terbuka set F1, ..., Fr, yang tersisa invarian oleh f (z) dan sedemikian rupa sehingga: persatuan Fi adalah padat dalam pesawat dan f (z) berperilaku dalam biasa dan cara yang sama pada masing-masing set Fi. Pernyataan terakhir berarti bahwa termini dari urutan iterasi yang dihasilkan oleh titik Fi yang baik justru set yang sama, yang kemudian siklus yang terbatas, atau mereka siklus terbatas melingkar atau annular berbentuk set yang berbohong konsentris. Dalam kasus pertama siklus ini menarik, di kedua adalah netral. set ini Fi adalah domain Fatou f (z), dan serikat mereka adalah Fatou set F (f) dari f (z). Setiap domain Fatou berisi setidaknya satu titik kritis f (z), yaitu, (terbatas) titik z memuaskan f '(z) = 0, atau z = ∞, jika tingkat pembilang p (z) setidaknya dua lebih besar dari tingkat q denominator (z), atau jika f (z) = 1 / g (z) + c untuk beberapa c dan fungsi rasional g (z) memenuhi kondisi ini. Komplemen dari F (f) adalah Julia set J (f) dari f (z). J (f) adalah satu set tempat padat (itu tanpa poin interior) dan set terhitung (dari kardinalitas yang sama dengan bilangan real). Seperti F (f), J (f) yang tersisa invarian oleh f (z), dan ini mengatur iterasi yang memukul mundur, yang berarti bahwa | f (z) - f (w) |> | z - w | untuk semua w di lingkungan z (dalam J (f)). Ini berarti bahwa f (z) berperilaku berantakan di set Julia. Meskipun ada poin di Julia mengatur urutan yang iterasi terbatas, hanya ada sejumlah dihitung dari titik-titik tersebut (dan mereka membuat bagian yang sangat kecil dari set Julia). Urutan yang dihasilkan oleh titik-titik di luar himpunan ini berperilaku berantakan, fenomena yang disebut kekacauan deterministik. Telah ada penelitian yang luas pada Fatou mengatur dan Julia set fungsi rasional iterasi, yang dikenal sebagai peta rasional. Misalnya, diketahui bahwa Fatou set peta rasional memiliki baik 0,1,2 atau jauh lebih banyak komponen. [4] Setiap komponen dari Fatou set peta rasional dapat diklasifikasikan ke dalam salah satu dari empat kelas yang berbeda. [ 5] deskripsi Setara dari Julia set [sunting] J (f) adalah himpunan tertutup terkecil yang mengandung setidaknya tiga titik yang benar-benar lain di bawah f. J (f) adalah penutupan himpunan memukul mundur poin periodik. Untuk semua tapi paling banyak dua poin z ∈ X, himpunan Julia adalah himpunan titik-titik batas orbit mundur penuh igcup_n f ^? {-} n (z). (Hal ini menunjukkan algoritma sederhana untuk merencanakan Julia set, lihat di bawah.) Jika f adalah seluruh fungsi, maka J (f) adalah batas himpunan titik-titik yang menyatu hingga tak terbatas di bawah iterasi. Jika f adalah polinomial, maka J (f) adalah batas diisi Julia set; yaitu, titik-titik yang orbitnya di bawah iterasi dari f tetap dibatasi. Sifat Julia set dan Fatou set [sunting] The Julia mengatur dan Fatou set f keduanya benar-benar lain di bawah iterasi dari fungsi holomorphic f: [6] f ^ {- 1} (J (f)) = f (J (f)) = J (f) f ^ {- 1} (F (f)) = f (F (f)) = F (f) Contoh [sunting] Untuk f (z) = z ^ {2} himpunan Julia adalah lingkaran satuan dan ini iterasi diberikan dengan menggandakan sudut (sebuah operasi yang kacau pada titik-titik yang argumen bukan fraksi rasional 2pi ). Ada dua Fatou domain: interior dan eksterior lingkaran, dengan iterasi terhadap 0 dan ∞, masing-masing. Untuk f (z) = z ^ {2} - 2 set Julia adalah segmen garis antara -2 dan 2. Ada satu Fatou domain: tidak pada segmen garis poin beralih menuju ∞. (Terlepas dari pergeseran dan skala dari domain, iterasi ini setara dengan xo 4 (x -. Frac {1} {2}) ^ {2} pada interval unit, yang umumnya digunakan sebagai contoh sistem kacau) Kedua fungsi ini dari bentuk z ^ 2 + c, di mana c adalah bilangan kompleks. Untuk iterasi seperti set Julia tidak secara umum kurva sederhana, tetapi fraktal, dan untuk beberapa nilai c dapat mengambil bentuk mengejutkan. Lihat gambar di bawah ini. Julia set (putih) untuk fungsi rasional terkait dengan metode Newton untuk f: z → z3-1. Mewarnai Fatou diatur sesuai dengan penarik (akar f) Untuk beberapa fungsi f (z) kita dapat mengatakan sebelumnya bahwa set Julia adalah fraktal dan tidak kurva sederhana. Hal ini karena hasil sebagai berikut pada iterasi dari fungsi rasional: Teorema. Setiap domain Fatou memiliki batas yang sama, yang akibatnya adalah set Julia. Ini berarti bahwa setiap titik dari himpunan Julia adalah titik akumulasi untuk setiap domain Fatou. Oleh karena itu, jika ada lebih dari dua domain Fatou, setiap titik dari himpunan Julia harus memiliki poin lebih dari dua set terbuka berbeda jauh dekat, dan ini berarti bahwa himpunan Julia tidak bisa menjadi kurva sederhana. Fenomena ini terjadi, misalnya, ketika f (z) adalah iterasi Newton untuk memecahkan persamaan zn = 1 untuk n> 2: f (z) = z - frac {f (z)} {f '(z)} = frac {1 + (n-1) z ^ n} {nz ^ {n-1}}. Gambar di kanan menunjukkan kasus n = 3. polinomial kuadrat [sunting] Sebuah sistem dinamik yang sangat populer kompleks diberikan oleh keluarga polinomial kuadrat, kasus khusus dari peta rasional. Polinomial kuadrat dapat dinyatakan sebagai f_c (z) = z ^ 2 + c dimana c adalah parameter yang rumit. Diisi Julia ditetapkan untuk fc, c = 1-φ φ mana adalah rasio emas Julia ditetapkan untuk fc, c = (φ -2) + (φ-1) i = -0,4 + 0.6i Julia ditetapkan untuk fc, c = 0.285 + 0i Julia ditetapkan untuk fc, c = 0.285 + 0.01i Julia ditetapkan untuk fc, c = 0,45 + 0.1428i Julia set untuk fc, c = -0.70176-0.3842i Julia ditetapkan untuk fc, c = -0.835-0.2321i Julia ditetapkan untuk fc, c = -0,8 + 0.156i A Julia mengatur rencana yang menunjukkan Julia set untuk nilai yang berbeda dari c; menyerupai Mandelbrot set Pesawat parameter polinomial kuadrat - yaitu, pesawat kemungkinan c-nilai - menimbulkan Mandelbrot set terkenal. Memang, Mandelbrot set didefinisikan sebagai himpunan semua c sehingga J (f_c) terhubung. Untuk parameter luar Mandelbrot set, set Julia adalah ruang Cantor: dalam hal ini kadang-kadang disebut sebagai Fatou debu. Dalam banyak kasus, Julia set c terlihat seperti Mandelbrot set di lingkungan cukup kecil c. Hal ini benar, khususnya, untuk apa yang disebut 'Misiurewicz' parameter, parameter yaitu c yang titik kritis adalah pra-periodik. Misalnya: Pada c = i, yang lebih pendek, kaki depan kaki depan, set Julia tampak seperti petir bercabang. Pada c = -2, ujung ekor runcing panjang, set Julia adalah segmen garis lurus. Dengan kata lain Julia set J (f_c) secara lokal serupa di seluruh poin Misiurewicz. [7] Contoh Julia set [sunting] f (z) = z2 + 0,279 f (z) = z3 + 0.400 f (z) = z4 + 0,484 f (z) = Z5 + 0,544 f (z) = z6 + 0,590 f (z) = Z7 + 0,626 f (z) = exp (z) - 0.65 f (z) = exp (z3) - 0,59 f (z ) = exp (z3) - 0,621 f (z) = z * exp (z) + 0,04 f (z) = z2 * exp (z) + 0,21 f (z) = z3 * exp (z) + 0,33 f (z ) = z4 * exp (Z) + 0,41 f (z) = Sqr [Sinh (z2)] + (0.065,0.122i) f (z) = [(z2 + z) / Ln (z)] + (0,268, 0.060i) Generalisasi [sunting] Definisi Julia dan Fatou set dengan mudah membawa ke kasus peta tertentu yang gambarnya mengandung domain mereka; terutama transendental fungsi meromorphic dan terbatas-jenis peta Adam Epstein. Julia set juga sering didefinisikan dalam studi dinamika di beberapa variabel kompleks. Potensi fungsi dan jumlah iterasi yang nyata [sunting] The Julia ditetapkan untuk f (z) = z ^ {2} adalah lingkaran satuan, dan pada domain Fatou luar, potensi fungsi φ (z) didefinisikan oleh φ (z) = log | z |. Garis ekipotensial untuk fungsi ini adalah lingkaran konsentris. Seperti | f (z) | = | z | ^ {2} kita memiliki varphi (z) = {k lim_ oinfty} frac {log | z_k |} {2 ^ k}, di mana z_k adalah urutan iterasi yang dihasilkan oleh z . Untuk iterasi f yang lebih umum (z) = z ^ 2 + c, telah membuktikan bahwa jika himpunan Julia terhubung (yaitu, jika c milik (biasa) Mandelbrot set), maka terdapat peta ψ biholomorphic antara luar Fatou domain dan luar lingkaran satuan sehingga | psi (f (z)) | = | psi (z) |. ^ {2} [8] Hal ini berarti bahwa fungsi potensial pada domain Fatou luar didefinisikan melalui korespondensi ini diberikan oleh: . varphi (z) = {k lim_ oinfty} frac {log | z_k |} {2 ^ k} Rumus ini memiliki arti juga jika set Julia tidak terhubung, sehingga kita semua dapat c mendefinisikan fungsi potensial pada domain Fatou mengandung ∞ dengan rumus ini. Untuk fungsi rasional f umum (z) sehingga ∞ adalah titik kritis dan titik tetap, yaitu, sehingga tingkat m dari pembilang setidaknya dua lebih besar dari tingkat n
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