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- https://math.libretexts.org/Bookshelves/Scientific_Computing_Simulations_and_Modeling/The_Mandelbrot_and_Julia_sets_Anatomy_(Demidov)/4%3A_Attractors%2C_repellers.../4.2%3A_Local_dynamics_at_a_fixed_pointThen iterations (or images) of a point (z o + ε ) in the vicinity of a fixed point z ∗ = f(z ∗ ) are z k = f ok (z ∗ + ε ) = z ∗ + λ k ε + O(ε 2 ) ~ z ∗ + |λ| k e ikφ ε. That is, if we put coordinate ...Then iterations (or images) of a point (z o + ε ) in the vicinity of a fixed point z ∗ = f(z ∗ ) are z k = f ok (z ∗ + ε ) = z ∗ + λ k ε + O(ε 2 ) ~ z ∗ + |λ| k e ikφ ε. That is, if we put coordinate origin to z ∗ , after every iteration point z k+1 is rotated by angle φ with respect to the previous position z k and its radius is scaled by |λ|. For |λ| < 1 all points in the vicinity of attractor z ∗ move smoothly to z ∗ . You can see "star" structures made by orbit of the critical point.
- https://math.libretexts.org/Bookshelves/Scientific_Computing_Simulations_and_Modeling/The_Mandelbrot_and_Julia_sets_Anatomy_(Demidov)/3%3A_The_Julia_set/3.3%3A_The_Fundamental_Dichotomy_for_Julia_setsWe take a circle l', which goes through z = c = 4 . Each point of l' has two preimages ±(z - c) ½ with the exeption of z = c, which has the only preimage z c . Therefore preimage of l' is the figure e...We take a circle l', which goes through z = c = 4 . Each point of l' has two preimages ±(z - c) ½ with the exeption of z = c, which has the only preimage z c . Therefore preimage of l' is the figure eight curve l. l' never contains z = c value, therefore each point of l' has two different preimages ±(z - c) ½ and l can not be a figure eight curve. In a similar way any closed curve outside the Julia set never is a figure eight curve and can not break the Julia set.
- https://math.libretexts.org/Bookshelves/Scientific_Computing_Simulations_and_Modeling/The_Mandelbrot_and_Julia_sets_Anatomy_(Demidov)/6%3A_Periodic_and_preperiodic_points
- https://math.libretexts.org/Bookshelves/Scientific_Computing_Simulations_and_Modeling/The_Mandelbrot_and_Julia_sets_Anatomy_(Demidov)/5%3A_Renormalization_theory
- https://math.libretexts.org/Bookshelves/Scientific_Computing_Simulations_and_Modeling/The_Mandelbrot_and_Julia_sets_Anatomy_(Demidov)/8%3A_Illustrations/8.3%3A_Distance_Estimator_algorithmsColor is "proportional" to log(dc), where dc is the approximate distance between the point c and the nearest point in the Mandelbrot set. The distance estimate for Julia sets is very close [1] to the ...Color is "proportional" to log(dc), where dc is the approximate distance between the point c and the nearest point in the Mandelbrot set. The distance estimate for Julia sets is very close [1] to the ratio |G|/|G'|, where G(z o ) = lim k→∞ log|z k |/n k |G'(z o )| = lim k→∞ |dz k /dz o | / (n k |z k |) . dz k /dz o = 2 k z k -1 ... z o . in the center of the picture at z o = 0) the distance estimate diverges therefore we get the central spot and its preimages.
- https://math.libretexts.org/Bookshelves/Scientific_Computing_Simulations_and_Modeling/The_Mandelbrot_and_Julia_sets_Anatomy_(Demidov)/2%3A_Iterations_of_quadratic_maps/2.3%3A_The_Mandelbrot%2C_Julia_and_Fatou_setsThe Mandelbrot set (M) is the set of all points c on complex plane (parameter space) such that iterations z n+1 = z n 2 + c do not go to infinity (the starting point z o = 0 will be discussed later). ...The Mandelbrot set (M) is the set of all points c on complex plane (parameter space) such that iterations z n+1 = z n 2 + c do not go to infinity (the starting point z o = 0 will be discussed later). This is the famous "Douady's rabbit". The "white" triangle shows the orbit star of attracting period-3 orbit f: z 1 → z 2 → z 3 → z 1 . This cycle lies in n = 3 components of the interior of the J-set.
- https://math.libretexts.org/Bookshelves/Scientific_Computing_Simulations_and_Modeling/The_Mandelbrot_and_Julia_sets_Anatomy_(Demidov)/3%3A_The_Julia_set/3.1%3A_The_Julia_set_symmetryAs since iterations of the points f c (z j ) do not go to an attractor too, therefore the Julia sets are invariant under f c . Therefore the J(0) set is the circle with the unit radius r = 1. The map ...As since iterations of the points f c (z j ) do not go to an attractor too, therefore the Julia sets are invariant under f c . Therefore the J(0) set is the circle with the unit radius r = 1. The map f o wraps twice the circle onto itself and is similar to the Sawtooth map. In accordance with (*) f c maps one half of the Julia set onto the whole set. You can test by hand that any merging point of three bulbs is a preimage of the unstable fixed point z 1 and these preimages are dense in the set.
- https://math.libretexts.org/Bookshelves/Scientific_Computing_Simulations_and_Modeling/The_Mandelbrot_and_Julia_sets_Anatomy_(Demidov)/5%3A_Renormalization_theory/5.4%3A_Polynomial-like_mapsA polynomial-like map of degree d is a holomorphic map (MathWorld) f: U' → U such that every point in U has exactly d preimages in U', where U, U' are open sets isomorphic to disc and U contains U' in...A polynomial-like map of degree d is a holomorphic map (MathWorld) f: U' → U such that every point in U has exactly d preimages in U', where U, U' are open sets isomorphic to disc and U contains U' in its interior. Let f is the restriction of polynomial P to a set U'. As V maps to U with degree one, hence, there are points in U' that map to V and come back to U' afterwards never leaving the set U.
- https://math.libretexts.org/Bookshelves/Scientific_Computing_Simulations_and_Modeling/The_Mandelbrot_and_Julia_sets_Anatomy_(Demidov)/5%3A_Renormalization_theory/5.7%3A__Shaggy_midgetsIn the vicinity of the M 3 midget the map f c o3 is renormalizable, therefore for c = A you see Cauliflower-like J(A) midget (compare it with the J(0.35) set to the right). Moreover for c = A the map ...In the vicinity of the M 3 midget the map f c o3 is renormalizable, therefore for c = A you see Cauliflower-like J(A) midget (compare it with the J(0.35) set to the right). Moreover for c = A the map f A o23 is renormalizable too, therefore we see the tiny black circle of the J(0) midget on the left picture and cardioid of the M 23 midget on the parameter plane (at the top of the page to the right). At last the point B lies in one more M 23 midget and the J(C) set below is pure Cantor dust.
- https://math.libretexts.org/Bookshelves/Scientific_Computing_Simulations_and_Modeling/The_Mandelbrot_and_Julia_sets_Anatomy_(Demidov)/4%3A_Attractors%2C_repellers...
- https://math.libretexts.org/Bookshelves/Scientific_Computing_Simulations_and_Modeling/The_Mandelbrot_and_Julia_sets_Anatomy_(Demidov)/4%3A_Attractors%2C_repellers.../4.8%3A_Parabollic_fixed_points_and_Siegel_diskas we have seen before, for c → 1/4 attracting and repelling fixed points merge together and make parabollic fixed point with multiplicator λ = 1 (in the center of the white square below). You can see...as we have seen before, for c → 1/4 attracting and repelling fixed points merge together and make parabollic fixed point with multiplicator λ = 1 (in the center of the white square below). You can see that white points of the orbit starting at z = 0 are attracted to this parabollic point. If λ = e 2π i x (where x is real irrational number) then in vicinity of fixed point for "almost all" x the mapping is equivalent to rotation by 2π x angle.
