scholarly journals Visualization of Mandelbrot and Julia Sets of Möbius Transformations

2021 ◽  
Vol 5 (3) ◽  
pp. 73
Author(s):  
Leah K. Mork ◽  
Darin J. Ulness
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Iteration Step ◽  
Möbius Transformation ◽  
Möbius Transformations ◽  
Fractal Sets ◽  
Mandelbrot Sets ◽  
Möbius Transforms ◽  
Mobius Transformations

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

scholarly journals Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers

2019 ◽  
Vol 3 (1) ◽  
pp. 6 ◽  
Author(s):  
Vance Blankers ◽  
Tristan Rendfrey ◽  
Aaron Shukert ◽  
Patrick Shipman
Keyword(s):  
Dynamical Systems ◽  
Natural Number ◽  
Hyperbolic Plane ◽  
Julia Sets ◽  
Number System ◽  
Mandelbrot Set ◽  
Complex Numbers ◽  
Hyperbolic Numbers ◽  
Fractal Sets ◽  
Mandelbrot Sets

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form x + τ y for x , y ∈ R , and τ 2 = 1 but τ ≠ ± 1 , are the natural number system in which to encode geometric properties of the Minkowski space R 1 , 1 . We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets.

Complex dynamics of Möbius semigroups

2012 ◽  
Vol 32 (6) ◽  
pp. 1889-1929 ◽  
Author(s):  
DAVID FRIED ◽  
SEBASTIAN M. MAROTTA ◽  
RICH STANKEWITZ
Keyword(s):  
Complex Dynamics ◽  
Riemann Sphere ◽  
Julia Sets ◽  
Fibonacci Sequence ◽  
Rational Functions ◽  
Möbius Transformations ◽  
Random Dynamics ◽  
Compare And Contrast ◽  
Möbius Groups ◽  
Mobius Transformations

AbstractWe study the dynamics of semigroups of Möbius transformations on the Riemann sphere, especially their Julia sets and attractors. This theory relates to the dynamics of rational functions, rational semigroups, and Möbius groups and we compare and contrast these theories. We particularly examine Caruso’s family of Möbius semigroups, based on a random dynamics variant of the Fibonacci sequence.

scholarly journals The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function

2021 ◽  
Vol 5 (3) ◽  
pp. 92
Author(s):  
Pavel Trojovský ◽  
K Venkatachalam
Keyword(s):  
Julia Sets ◽  
Catalan Numbers ◽  
Iteration Step ◽  
Möbius Transformation ◽  
Iterated Function ◽  
Mobius Transformation

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function ηλ(z)=z2+λ. Their generalization was based on the composition of ηλ with the Möbius transformation μ(z)=1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of μ(ηλ(z)) (at z=0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation.

Conformal trace theorem for Julia sets of quadratic polynomials

2017 ◽  
Vol 39 (9) ◽  
pp. 2481-2506 ◽  
Author(s):  
A. CONNES ◽  
E. MCDONALD ◽  
F. SUKOCHEV ◽  
D. ZANIN
Keyword(s):  
Hausdorff Dimension ◽  
Jordan Curve ◽  
Hausdorff Measure ◽  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Trace Theorem ◽  
Quadratic Polynomials ◽  
Full Proof

If $c$ is in the main cardioid of the Mandelbrot set, then the Julia set $J$ of the map $\unicode[STIX]{x1D719}_{c}:z\mapsto z^{2}+c$ is a Jordan curve of Hausdorff dimension $p\in [1,2)$. We provide a full proof of a formula for the Hausdorff measure on $J$ in terms of singular traces announced by the first named author in 1996.

scholarly journals Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Sunhong Lee ◽  
Hyun Chol Lee ◽  
Mi Ran Lee ◽  
Seungpil Jeong ◽  
Gwang-Il Kim
Keyword(s):  
Complex Plane ◽  
Hermite Interpolation ◽  
Möbius Transformation ◽  
Möbius Transformations ◽  
Interpolation Problems ◽  
Pythagorean Hodograph ◽  
Improved Stability ◽  
Mobius Transformation ◽  
Extra Parameter ◽  
Mobius Transformations

We present an algorithm forC1Hermite interpolation using Möbius transformations of planar polynomial Pythagoreanhodograph (PH) cubics. In general, with PH cubics, we cannot solveC1Hermite interpolation problems, since their lack of parameters makes the problems overdetermined. In this paper, we show that, for each Möbius transformation, we can introduce anextra parameterdetermined by the transformation, with which we can reduce them to the problems determining PH cubics in the complex planeℂ. Möbius transformations preserve the PH property of PH curves and are biholomorphic. Thus the interpolants obtained by this algorithm are also PH and preserve the topology of PH cubics. We present a condition to be met by a Hermite dataset, in order for the corresponding interpolant to be simple or to be a loop. We demonstrate the improved stability of these new interpolants compared with PH quintics.

Accumulation theorems for quadratic polynomials

1996 ◽  
Vol 16 (3) ◽  
pp. 555-590 ◽  
Author(s):  
Dan Erik Krarup Sørensen
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Quadratic Polynomials ◽  
Hyperbolic Component ◽  
Non Local ◽  
External Ray ◽  
The One ◽  
Image Position ◽  
Symmetrical Points

AbstractWe consider the one-parameter family of quadratic polynomials:i.e. monic centered quadratic polynomials with an indifferent fixed point αtand prefixed point −αt. LetAt, be any one of the sets {0, ±αt}, {±αt}, {0, αt}, or {0, −αt}. Then we prove that for quadratic Julia sets corresponding to aGδ-dense subset ofthere is an explicitly given external ray accumulating onAt. In the caseAt= {±αt} the theorem is known as theDouady accumulation theorem.Corollaries are the non-local connectivity of these Julia sets and the fact that all such Julia sets contain a Cremer point. Existence of non-locally connected quadratic Julia sets of Hausdorff dimension two is derived by using a recent result of Shishikura. By tuning, the results hold on the boundary of any hyperbolic component of the Mandelbrot set.Finally, we concentrate on quadratic Cremer point polynomials. Here we prove that any ray accumulating on two symmetrical points of the Julia set must accumulate the origin. As a consequence, the denseGδsets arising from the first two possible choices ofAtare the same. We also prove that, if two distinct rays accumulate both to two distinct points, then the rays must accumulate on a common continuum joining the two points. This supports the conjecture that αtand –αtmay be joined by an arc in the Julia set.

DYNAMIC ANALYSIS OF THE CAROTID–KUNDALINI MAP

2008 ◽  
Vol 22 (04) ◽  
pp. 243-262 ◽  
Author(s):  
XINGYUAN WANG ◽  
QINGYONG LIANG ◽  
JUAN MENG
Keyword(s):  
Analytic Function ◽  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Quantitative Criterion ◽  
Boundary Equation ◽  
Computer Aided ◽  
Definition Of ◽  
Mathematics Method ◽  
Periodic Bifurcation

The nature of the fixed points of the Carotid–Kundalini (C–K) map was studied and the boundary equation of the first bifurcation of the C–K map in the parameter plane is presented. Using the quantitative criterion and rule of chaotic system, the paper reveals the general features of the C–K Map transforming from regularity to chaos. The following conclusions are obtained: (i) chaotic patterns of the C–K map may emerge out of double-periodic bifurcation; (ii) the chaotic crisis phenomena are found. At the same time, the authors analyzed the orbit of critical point of the complex C–K Map and put forward the definition of Mandelbrot–Julia set of the complex C–K Map. The authors generalized the Welstead and Cromer's periodic scanning technique and using this technology constructed a series of the Mandelbrot–Julia sets of the complex C–K Map. Based on the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, we investigated the symmetry of the Mandelbrot–Julia set and studied the topological inflexibility of distribution of the periodic region in the Mandelbrot set, and found that the Mandelbrot set contains abundant information of the structure of Julia sets by finding the whole portray of Julia sets based on Mandelbrot set qualitatively.

scholarly journals Total disconnectedness of Julia sets of random quadratic polynomials

2021 ◽  
pp. 1-17
Author(s):  
KRZYSZTOF LECH ◽  
ANNA ZDUNIK
Keyword(s):  
Dynamical System ◽  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Quadratic Polynomials ◽  
Large Disc ◽  
Fatou Sets ◽  
Autonomous Version ◽  
Composition Of Functions ◽  
Totally Disconnected

Abstract For a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

scholarly journals Fractals Parrondo’s Paradox in Alternated Superior Complex System

2021 ◽  
Vol 5 (2) ◽  
pp. 39
Author(s):  
Yi Zhang ◽  
Da Wang
Keyword(s):  
Complex System ◽  
Julia Set ◽  
Julia Sets ◽  
Time Algorithm ◽  
Mandelbrot Set ◽  
Escape Time ◽  
Parrondo’S Paradox ◽  
The One ◽  
Totally Disconnected ◽  
Superior Mandelbrot Set

This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria.

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