991 results on '"Julia set"'
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2. Analysis of Neutrosophic Set, Julia Set in Aircraft Crash
- Author
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M.N. Bharathi and G. Jayalalitha
- Subjects
neutrosophicset ,neutrosophic fuzzy set ,fractals ,julia set ,bird collision ,Mathematics ,QA1-939 ,Electronic computers. Computer science ,QA75.5-76.95 - Abstract
A neutrosophic fuzzy set that generalizes the classical set is represented by a closed interval [0,1]. Let us generalize the fuzzy set to the Neutrosophic set, which is defined as three membership functions between interval ]-0,1+ [. This paper examines the bird collision problem within a more grounded Neutrosophic framework. The collision between the bird and the aircraftis represented in the Neutrosophic domain as a stationary point because it occurs at a distance too great for any other auditory signal to discern. Numerous methods exist for solving the airplane problem, including a Julia set in complex numbers. Bird collisions resulting from aircraft crashes add complication. An aviation signalis utilized to avoid anaccident in a bird strike or bird collision.
- Published
- 2024
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3. Public Art Design Practice under Visual Communication Design
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Huang Pu, Hao Wei, and Jin Qiuyue
- Subjects
visual communication design ,fractal algorithm ,escape time ,julia set ,public art design ,97u10 ,Mathematics ,QA1-939 - Abstract
The expansion of visual communication design in public art makes the scope of visual communication wide. This new art form reflects the cross and integration of disciplines but also makes the form of public art in our life richer, and people can get more beautiful enjoyment. Research on the fractal algorithm in the escape time algorithm studied the Julia set in different function conditions of the fractal graph of the change and the function of different indices of the joint. The fractal graph obtained a colorful, more tense structure. Subsequently, it is applied to public art design, and after testing its performance, the fractal design of visual communication is combined with the IPA model to explore the practical effect of the fractal design in public art design. The results show that the improved fractal algorithm proposed in this paper increases the pattern generation rate from 61.5% to 92.6%. The fractal dimension measurement of 15 typical batik patterns shows that more than 85% of the batik patterns have an average value of more than 15000.
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- 2024
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4. The lower bound on the measure of sets consisting of Julia limiting directions of solutions to some complex equations associated with Petrenko's deviation
- Author
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Guowei Zhang
- Subjects
petrenko's deviation ,julia set ,entire function ,complex equation ,Mathematics ,QA1-939 - Abstract
In the value distribution theory of complex analysis, Petrenko's deviation is to describe more precisely the quantitative relationship between $ T (r, f) $ and $ \log M (r, f) $ when the modulus of variable $ |z| = r $ is sufficiently large. In this paper we introduce Petrenko's deviations to the coefficients of three types of complex equations, which include difference equations, differential equations and differential-difference equations. Under different assumptions we study the lower bound of limiting directions of Julia sets of solutions of these equations, where Julia set is an important concept in complex dynamical systems. The results of this article show that the lower bound of limiting directions mentioned above is closely related to Petrenko's deviation, and our conclusions improve some known results.
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- 2023
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5. Exploring parameter spaces in complex dynamics
- Author
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Pedro Iván Suárez Navarro
- Subjects
complex dynamics ,blaschke products ,mandelbrot set ,julia set ,Applied mathematics. Quantitative methods ,T57-57.97 ,Mathematics ,QA1-939 - Abstract
We show the structure of the parameter space for a family of rational maps containing Blaschke products. Through numerical simulations using the orbit of a single critical point, we reveal the existence of infinitely many Mandelbrot-like sets along the unit circle, as well as eight-like structures in other regions of parameter space. We pose some open questions related to the parameter space of these functions.
- Published
- 2023
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6. Singular Perturbations of Multibrot Set Polynomials
- Author
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Figen Çilingir
- Subjects
newton basin ,rational iteration ,julia set ,perturbation ,Mathematics ,QA1-939 - Abstract
We will give a complete description of the dynamics of the rational map $N_{F_{M_c}}(z)=\frac{3z^4-2z^3+c}{4z^3-3z^2+c}$ where c is a complex parameter. These are rational maps $N_{F_{M_c}}$ arising from Newton's method. The polynomial of Newton iteration function is obtained from singularly perturbed of the Multibrot set polynomial.
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- 2022
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7. Synchronization of Julia Sets in Three-Dimensional Discrete Financial Models
- Author
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Zhongyuan Zhao, Yongping Zhang, and Dadong Tian
- Subjects
financial model ,Julia set ,synchronization ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
When aiming to achieve consistency in fractal characteristics between different models, it is crucial to consider the synchronization of Julia sets. This paper studies the synchronization of Julia sets in three-dimensional discrete financial models. First, three-dimensional discrete financial models with different model parameters are proposed and their Julia sets are presented. According to the model forms, two kinds of synchronous couplers that can achieve synchronization of Julia sets between different models are designed by changing the synchronization parameters. The proposed synchronization method is theoretically derived and the efficiency of different synchronous couplers are compared. Finally, the effectiveness is verified by Julia sets graphics. This method has reference value for theoretical research into financial models in the field of fractals.
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- 2023
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8. Difference Equations and Julia Sets of Several Functions for Degenerate q-Sigmoid Polynomials
- Author
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Jung-Yoog Kang and Cheon-Seoung Ryoo
- Subjects
(q,h)-derivative ,(q,h)-difference equations ,DQS polynomials ,Julia set ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
In this article, we construct a new type of degenerate q-sigmoid (DQS) polynomial for sigmoid functions containing quantum numbers and find several difference equations related to it. We check how each point moves by iteratively synthesizing a quartic degenerate q-sigmoid (DQS) polynomial that appears differently depending on q in the space of a complex structure. We also construct Julia sets associated with quartic DQS polynomials and find their features. Based on this, we make some conjectures.
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- 2023
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9. Mandelbrot and Julia Sets of Transcendental Functions Using Picard–Thakur Iteration
- Author
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Ashish Bhoria, Anju Panwar, and Mohammad Sajid
- Subjects
algorithms ,fractals ,Julia set ,Mandelbrot set ,Picard–Thakur iteration ,escape criterion ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
The majority of fractals’ dynamical behavior is determined by escape criteria, which utilize various iterative procedures. In the context of the Julia and Mandelbrot sets, the concept of “escape” is a fundamental principle used to determine whether a point in the complex plane belongs to the set or not. In this article, the fractals of higher importance, i.e., Julia sets and Mandelbrot sets, are visualized using the Picard–Thakur iterative procedure (as one of iterative methods) for the complex sine Tc(z)=asin(zr)+bz+c and complex exponential Tc(z)=aezr+bz+c functions. In order to obtain the fixed point of a complex-valued sine and exponential function, our concern is to use the fewest number of iterations possible. Using MATHEMATICA 13.0, some enticing and intriguing fractals are generated, and their behavior is then illustrated using graphical examples; this is achieved depending on the iteration parameters, the parameters ‘a’ and ‘b’, and the parameters involved in the series expansion of the sine and exponential functions.
- Published
- 2023
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10. A study of a meromorphic perturbation of the sine family
- Author
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Domínguez Patricia and Vázquez Josué
- Subjects
iteration ,fixed points ,fatou set ,julia set ,37f10 ,30d05 ,Mathematics ,QA1-939 - Abstract
We study the dynamics of a meromorphic perturbation of the family λsinz\lambda \sin z by adding a pole at zero and a parameter μ\mu , that is, fλ,μ(z)=λsinz+μ/z{f}_{\lambda ,\mu }\left(z)=\lambda \sin z+\mu \hspace{-0.08em}\text{/}\hspace{-0.08em}z, where λ,μ∈C⧹{0}\lambda ,\mu \in {\mathbb{C}}\hspace{-0.16em}\setminus \hspace{-0.16em}\left\{0\right\}. We study some geometrical properties of fλ,μ{f}_{\lambda ,\mu } and prove that the imaginary axis is invariant under fn{f}^{n} and belongs to the Julia set when ∣λ∣≥1| \lambda | \ge 1. We give a set of parameters (λ,μ)\left(\lambda ,\mu ), such that the Fatou set of fλ,μ{f}_{\lambda ,\mu } has two super-attracting domains. If λ=1\lambda =1 and μ∈(0,2)\mu \in \left(0,2), the Fatou set of f1,μ{f}_{1,\mu } has two attracting domains. Also, we give parameters λ,μ\lambda ,\mu such that ±π/2\pm \pi \hspace{-0.08em}\text{/}\hspace{-0.08em}2 are fixed points of fλ,μ{f}_{\lambda ,\mu } and the Fatou set of fλ,μ{f}_{\lambda ,\mu } contains attracting domains, parabolic domains, and Siegel discs, we present examples of these domains. This paper closes with an example of fλ,μ{f}_{\lambda ,\mu }, where the Fatou set contains two types of domains, for λ,μ\lambda ,\mu given.
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- 2022
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11. The Hausdorff dimension of the Julia sets concerning generated renormalization transformation
- Author
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Tingting Li and Junyang Gao
- Subjects
renormlization transformation ,potts model ,asymptotic formula ,hausdorff dimension ,julia set ,Mathematics ,QA1-939 - Abstract
Considering a family of rational map $ {U_{mn\lambda }} $ of the renormalization transformation of the generalized diamond hierarchical Potts model, we give the asymptotic formula of the Hausdorff dimension of the Julia sets of $ {U_{mn\lambda }} $ as the parameter $ \lambda $ tends to infinity, here $ {U_{mn\lambda }} = {\left[ {\frac{{{{\left( {z + \lambda - 1} \right)}^n} + \left( {\lambda - 1} \right){{\left( {z - 1} \right)}^n}}}{{{{\left( {z + \lambda - 1} \right)}^n} - {{\left( {z - 1} \right)}^n}}}} \right]^m}, $ where $ m \ge 2 $, $ n \ge 2 $ are two natural numbers, $ \lambda \in {{\mathbb{C}} } $.
- Published
- 2022
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12. The Study Geometry Fractals Designed on Batik Motives
- Author
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Juhari Juhari
- Subjects
fractals ,julia set ,seirpinski ,Mathematics ,QA1-939 - Abstract
This research was conducted to gain some patterns of fractals Julia set and Seirpinski that applied on Batik then creating Batik that has many varied motives and multifaceted. There are three steps in formulating the patterns of fractals of Julia set and Seirpinski. First, build the fractals by analyzing the function of fractals Julia set and determine the plane’s coordinate which you want to use. In this case, we use square and rectangle which will be created by using fractals patterns Seirpinski. Second, create a batik motives from fractals pattern Julia set and Seirpinski by using geometry transformation. The geometry transformation which will be used are translation, dilatation, reflection, and rotation. The last, combine some batik motives which were created by using image processing. It was summation of two images processing. The result is batik motives that has many variated and multifaceted.
- Published
- 2019
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13. A Four Step Feedback Iteration and Its Applications in Fractals
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Asifa Tassaddiq, Muhammad Tanveer, Muhammad Azhar, Waqas Nazeer, and Sania Qureshi
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imaging ,complex function ,Julia set ,Mandelbrot set ,multi-corn ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
Fractals play a vital role in modeling the natural environment. The present aim is to investigate the escape criterion to generate specific fractals such as Julia sets, Mandelbrot sets and Multi-corns via F-iteration using complex functions h(z)=zn+c, h(z)=sin(zn)+c and h(z)=ezn+c, n≥2,c∈C. We observed some beautiful Julia sets, Mandelbrot sets and Multi-corns for n = 2, 3 and 4. We generalize the algorithms of the Julia set and Mandelbrot set to visualize some Julia sets, Mandelbrot sets and Multi-corns. Moreover, we calculate image generation time in seconds at different values of input parameters.
- Published
- 2022
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14. A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative Scheme
- Author
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Darshana J. Prajapati, Shivam Rawat, Anita Tomar, Mohammad Sajid, and R. C. Dimri
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fixed point ,Mann orbit ,Julia set ,bifurcation ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
In this research, we look at the Julia set patterns that are linked to the entire transcendental function f(z)=aezn+bz+c, where a,b,c∈C and n≥2, using the Mann iterative scheme, and discuss their dynamical behavior. The sophisticated orbit structure of this function, whose Julia set encompasses the entire complex plane, is described using symbolic dynamics. We also present bifurcation diagrams of Julia sets generated using the proposed iteration and function, which altogether contain four parameters, and discuss the graphical analysis of bifurcation occurring in the family of this function.
- Published
- 2022
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15. Properties of q-Differential Equations of Higher Order and Visualization of Fractal Using q-Bernoulli Polynomials
- Author
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Cheon-Seoung Ryoo and Jung-Yoog Kang
- Subjects
q-Bernoulli polynomials ,q-difference equation of higher order ,Mandelbrot set ,Julia set ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
We introduce several q-differential equations of higher order which are related to q-Bernoulli polynomials and obtain a symmetric property of q-differential equations of higher order in this paper. By giving q-varying variations, we identify the shape of the approximate roots of q-Bernoulli polynomials, a solution of q-differential equations of higher order, and find several conjectures associated with them. Furthermore, based on q-Bernoulli polynomials, we create a Mandelbrot set and a Julia set to find a variety of related figures.
- Published
- 2022
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16. Consensus of Julia Sets
- Author
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Weihua Sun and Shutang Liu
- Subjects
Julia set ,consensus ,synchronization ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
The Julia set is one of the most important sets in fractal theory. The previous studies on Julia sets mainly focused on the properties and graph of a single Julia set. In this paper, activated by the consensus of multi-agent systems, the consensus of Julia sets is introduced. Moreover, two types of the consensus of Julia sets are proposed: one is with a leader and the other is with no leaders. Then, controllers are designed to achieve the consensus of Julia sets. The consensus of Julia sets allows multiple different Julia sets to be coupled. In practical applications, the consensus of Julia sets provides a tool to study the consensus of group behaviors depicted by a Julia set. The simulations illustrate the efficacy of these methods.
- Published
- 2022
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17. Computer research of the holomorphic dynamics of exponential and linear-exponential maps
- Author
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I. V. Matyushkin and Mariya Andreevna Zapletina
- Subjects
experimental mathematics ,bifurcation ,fractal ,holomorphic dynamics ,complex maps ,nonlinear dynamics ,Julia set ,Fatou set ,complex-valued exponent ,Applied mathematics. Quantitative methods ,T57-57.97 ,Mathematics ,QA1-939 - Abstract
The work belongs to the direction of experimental mathematics, which investigates the properties of mathematical objects by the computing facilities of a computer. The base is an exponential map, its topological properties (Cantor's bouquets) differ from properties of polynomial and rational complex-valued functions. The subject of the study are the character and features of the Fatou and Julia sets, as well as the equilibrium points and orbits of the zero of three iterated complex-valued mappings: $f:z \to (1+ \mu) \exp (iz)$, $g : z \to \big(1+ \mu |z - z^*|\big) \exp (iz)$, $h : z \to \big(1+ \mu (z - z^* )\big) \exp (iz)$, with $z,\mu \in \mathbb{C}$, $z^* : \exp (iz^*) = z^*$. For a quasilinear map g having no analyticity characteristic, two bifurcation transitions were discovered: the creation of a new equilibrium point (for which the critical value of the linear parameter was found and the bifurcation consists of "fork" type and "saddle"-node transition) and the transition to the radical transformation of the Fatou set. A nontrivial character of convergence to a fixed point is revealed, which is associated with the appearance of "valleys" on the graph of convergence rates. For two other maps, the monoperiodicity of regimes is significant, the phenomenon of "period doubling" is noted (in one case along the path $39\to 3$, in the other along the path $17\to 2$), and the coincidence of the period multiplicity and the number of sleeves of the Julia spiral in a neighborhood of a fixed point is found. A rich illustrative material, numerical results of experiments and summary tables reflecting the parametric dependence of maps are given. Some questions are formulated in the paper for further research using traditional mathematics methods.
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- 2018
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18. Phenomenon of Scattering of Zeros of the (p,q)-Cosine Sigmoid Polynomials and (p,q)-Sine Sigmoid Polynomials
- Author
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Cheon Seoung Ryoo and Jung Yoog Kang
- Subjects
(p,q)-cosine sigmoid polynomials ,(p,q)-sine sigmoid polynomials ,approximate roots ,Mandelbrot set ,Julia set ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
In this paper, we define (p,q)-cosine and sine sigmoid polynomials. Based on this, the properties of each polynomial, and the structure and assumptions of its roots, can be identified. Properties can also be determined by the changes in p and q.
- Published
- 2021
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19. The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function
- Author
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Pavel Trojovský and K Venkatachalam
- Subjects
fractal ,Mandelbrot set ,Julia set ,Möbius transformation ,iterated function ,Catalan numbers ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
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.
- Published
- 2021
- Full Text
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20. Visualization of Mandelbrot and Julia Sets of Möbius Transformations
- Author
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Leah K. Mork and Darin J. Ulness
- Subjects
fractal ,Mandelbrot set ,Julia set ,Möbius transformation ,hyperbolic geometry ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
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.
- Published
- 2021
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21. Fractals Parrondo’s Paradox in Alternated Superior Complex System
- Author
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Yi Zhang and Da Wang
- Subjects
Parrodo’paradox ,Mann iteration ,Julia set ,alternated system ,connectivity ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
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.
- Published
- 2021
- Full Text
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22. Aplikasi Himpunan Julia dalam Membuat Rancangan Motif Fraktal Songket Palembang
- Author
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Eka Susanti
- Subjects
Fractal Motif ,Julia Set ,Sadum Tumpal Ulos Motif ,Mathematics ,QA1-939 - Abstract
Songket is one of the Indonesian arts and culture into community characteristics Palembang. Palembang songket motifs influenced by Islamic culture, in general, Palembang songket motifs shaped flowers. In this paper are given motif fractal Palembang songket motifs Sadum Tumpal Ulos. This motif is formed from the set of Julia with , with z = a + ib and n = 5.5; n = 6 and n = z.
- Published
- 2016
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23. The Hausdorff dimension of the Julia sets concerning generated renormalization transformation
- Author
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Junyang Gao and Tingting Li
- Subjects
asymptotic formula ,Pure mathematics ,renormlization transformation ,General Mathematics ,Julia set ,Renormalization ,Transformation (function) ,Hausdorff dimension ,julia set ,potts model ,QA1-939 ,hausdorff dimension ,Mathematics - Abstract
Considering a family of rational map $ {U_{mn\lambda }} $ of the renormalization transformation of the generalized diamond hierarchical Potts model, we give the asymptotic formula of the Hausdorff dimension of the Julia sets of $ {U_{mn\lambda }} $ as the parameter $ \lambda $ tends to infinity, here \begin{document}$ {U_{mn\lambda }} = {\left[ {\frac{{{{\left( {z + \lambda - 1} \right)}^n} + \left( {\lambda - 1} \right){{\left( {z - 1} \right)}^n}}}{{{{\left( {z + \lambda - 1} \right)}^n} - {{\left( {z - 1} \right)}^n}}}} \right]^m}, $\end{document} where $ m \ge 2 $, $ n \ge 2 $ are two natural numbers, $ \lambda \in {{\mathbb{C}} } $.
- Published
- 2022
24. Bers slices in families of univalent maps
- Author
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Sabyasachi Mukherjee, Nikolai Makarov, and Kirill Lazebnik
- Subjects
Polynomial ,Mathematics - Complex Variables ,Mathematics::Complex Variables ,General Mathematics ,Image (category theory) ,Closure (topology) ,30C10, 37F10 ,Dynamical Systems (math.DS) ,Julia set ,Combinatorics ,Reflection (mathematics) ,FOS: Mathematics ,Embedding ,Ideal (ring theory) ,Mathematics - Dynamical Systems ,Complex Variables (math.CV) ,Reflection group ,Mathematics - Abstract
We construct embeddings of Bers slices of ideal polygon reflection groups into the classical family of univalent functions $\Sigma$. This embedding is such that the conformal mating of the reflection group with the anti-holomorphic polynomial $z\mapsto\overline{z}^d$ is the Schwarz reflection map arising from the corresponding map in $\Sigma$. We characterize the image of this embedding in $\Sigma$ as a family of univalent rational maps. Moreover, we show that the limit set of every Kleinian reflection group in the closure of the Bers slice is naturally homeomorphic to the Julia set of an anti-holomorphic polynomial., Comment: Figure 1 added to illustrate the main result, and some other minor changes to the introduction
- Published
- 2021
25. Dynamics of the meromorphic families $f_\lambda=\lambda \tan^pz^q$
- Author
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Linda Keen and Tao Chen
- Subjects
Essential singularity ,Pure mathematics ,Algebra and Number Theory ,Dense set ,Applied Mathematics ,Fixed point ,Lambda ,Julia set ,Cantor set ,Singular value ,Geometry and Topology ,Analysis ,Mathematics ,Meromorphic function - Abstract
This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. Here we look at a generalization of the family of polynomials $P_a(z)=z^{d-1}(z- \frac{da}{(d-1)})$, the family $f_{\lambda}=\lambda \tan^p z^q$. These functions have a super-attractive fixed point, and, depending on $p$, one or two asymptotic values. Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here. Adding transcendental methods to standard ones, we give a description of the dynamical properties; in particular we prove the Julia set of a hyperbolic map is either connected and locally connected or a Cantor set. We also give a description of the parameter plane of the family $f_{\lambda}$. Again there are similarities to and differences from the parameter plane of the family $P_a$ and again there are new techniques. In particular, we prove there is dense set of points on the boundaries of the hyperbolic components that are accessible along curves and we characterize these points.
- Published
- 2021
26. Cutpoints of Invariant Subcontinua of Polynomial Julia Sets
- Author
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Vladlen Timorin, Alexander Blokh, and Lex Oversteegen
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Combinatorics ,Polynomial (hyperelastic model) ,Riemann hypothesis ,symbols.namesake ,Plane (geometry) ,General Mathematics ,symbols ,Branched covering ,Fixed point ,Invariant (mathematics) ,Complex quadratic polynomial ,Julia set ,Mathematics - Abstract
We prove fixed point results for branched covering maps f of the plane. For complex polynomials P with Julia set $$J_{P}$$ these imply that periodic cutpoints of some invariant subcontinua of $$J_{P}$$ are also cutpoints of $$J_{P}$$ . We deduce that, under certain assumptions on invariant subcontinua Q of $$J_{P}$$ , every Riemann ray to Q landing at a periodic repelling/parabolic point $$x\in Q$$ is isotopic to a Riemann ray to $$J_{P}$$ relative to Q.
- Published
- 2021
27. The Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets
- Author
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Abdul Aziz Shahid, Waqas Nazeer, and Krzysztof Gdawiec
- Subjects
Discrete mathematics ,Degree (graph theory) ,Picard–Mann iteration ,General Mathematics ,010102 general mathematics ,Mann iteration ,Julia set ,Fixed-point theorem ,escape criterion ,Mandelbrot set ,01 natural sciences ,Convexity ,010101 applied mathematics ,Fractal ,0101 mathematics ,Complex polynomial ,Mathematics - Abstract
In recent years, researchers have studied the use of different iteration processes from fixed point theory in the generation of complex fractals. For instance, the Mann, Ishikawa, Noor, Jungck–Mann and Jungck–Ishikawa iterations have been used. In this paper, we study the use of the Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets. We prove the escape criterion for the $$(k+1)$$ ( k + 1 ) st degree complex polynomial. Moreover, we present some graphical and numerical examples regarding Mandelbrot and Julia sets generated using the proposed iteration.
- Published
- 2021
28. Variasi Motif Batik Palembang Menggunakan Sistem Fungsi Teriterasi dan Himpunan Julia
- Author
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Eka Susanti
- Subjects
Iterated Function Systems ,IFS ,Julia Set ,Carpet Sierpinski ,Mathematics ,QA1-939 - Abstract
Batik Palembang has a distinctive design with bright colour. There are several design, in this research, can be visualized songket batik design and jumputan batik design. Jumputan and songket batik design with the help of the software can be visualized using iterated function systems (IFS) and the Julia set. Songket batik design can be visualized with a combination of Sierpinski Carpet and Julia set while jumputan battik design can be visualized with a combination of some of the Julia set.
- Published
- 2015
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29. The Arakelov-Zhang pairing and Julia sets
- Author
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Andrew Bridy and Matthew H. Larson
- Subjects
Combinatorics ,Simple (abstract algebra) ,Applied Mathematics ,General Mathematics ,Pairing ,Term (logic) ,Algebraic number field ,Space (mathematics) ,Julia set ,Measure (mathematics) ,Upper and lower bounds ,Mathematics - Abstract
The Arakelov-Zhang pairing ⟨ ψ , ϕ ⟩ \langle \psi ,\phi \rangle is a measure of the “dynamical distance” between two rational maps ψ \psi and ϕ \phi defined over a number field K K . It is defined in terms of local integrals on Berkovich space at each completion of K K . We obtain a simple expression for the important case of the pairing with a power map, written in terms of integrals over Julia sets. Under certain disjointness conditions on Julia sets, our expression simplifies to a single canonical height term; in general, this term is a lower bound. As applications of our method, we give bounds on the difference between the canonical height h ϕ h_\phi and the standard Weil height h h , and we prove a rigidity statement about polynomials that satisfy a strong form of good reduction.
- Published
- 2021
30. Combinatorics of criniferous entire maps with escaping critical values
- Author
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Leticia Pardo-Simón
- Subjects
medicine.medical_specialty ,Class (set theory) ,Pure mathematics ,Mathematics - Complex Variables ,Entire function ,media_common.quotation_subject ,010102 general mathematics ,Escaping set ,Topological dynamics ,Dynamical Systems (math.DS) ,Function (mathematics) ,Infinity ,01 natural sciences ,Julia set ,010101 applied mathematics ,Bounded function ,FOS: Mathematics ,medicine ,Geometry and Topology ,Mathematics - Dynamical Systems ,Complex Variables (math.CV) ,0101 mathematics ,Mathematics ,media_common - Abstract
A transcendental entire function is called criniferous if every point in its escaping set can eventually be connected to infinity by a curve of escaping points. Many transcendental entire functions with bounded singular set have this property, and this class has recently attracted much attention in complex dynamics. In the presence of escaping critical values, these curves break or split at (preimages of) critical points. In this paper, we develop combinatorial tools that allow us to provide a complete description of the escaping set of any criniferous function without asymptotic values on its Julia set. In particular, our description precisely reflects the splitting phenomenon. This combinatorial structure provides the foundation for further study of this class of functions. For example, we use these results in [arXiv:1905.03778] to give the first full description of the topological dynamics of a class of transcendental entire maps with unbounded postsingular set., Comment: 28 pages, 3 figures. Refined results of those in sections 2,3 and 4 of the second version of arXiv:1905.03778
- Published
- 2021
31. Quasisymmetric uniformization and Hausdorff dimensions of Cantor circle Julia sets
- Author
-
Wei-Yuan Qiu and Fei Yang
- Subjects
Mathematics::Dynamical Systems ,Degree (graph theory) ,Mathematics::Complex Variables ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hausdorff space ,Mathematics::General Topology ,01 natural sciences ,Julia set ,Infimum and supremum ,Moduli space ,Conformal dimension ,Combinatorics ,Hausdorff dimension ,0101 mathematics ,Uniformization (set theory) ,Mathematics - Abstract
For Cantor circle Julia sets of hyperbolic rational maps, we prove that they are quasisymmetrically equivalent to standard Cantor circles (i.e., connected components are round circles). This gives a quasisymmetric uniformization of all Cantor circle Julia sets of hyperbolic rational maps. By analyzing the combinatorial information of the rational maps whose Julia sets are Cantor circles, we give a computational formula of the number of the Cantor circle hyperbolic components in the moduli space of rational maps for any fixed degree. We calculate the Hausdorff dimensions of the Julia sets which are Cantor circles, and prove that for any Cantor circle hyperbolic component $\mathcal{H}$ in the space of rational maps, the infimum of the Hausdorff dimensions of the Julia sets of the maps in $\mathcal{H}$ is equal to the conformal dimension of the Julia set of any representative $f_0\in\mathcal{H}$, and that the supremum of the Hausdorff dimensions is equal to $2$.
- Published
- 2021
32. Stochastic adding machines based on Bratteli diagrams
- Author
-
Glauco Valle, Danilo Antonio Caprio, Ali Messaoudi, Universidade Estadual Paulista (UNESP), Universidade Federal do Rio de Janeiro (UFRJ), and Universidade Estadual Paulista (Unesp)
- Subjects
Adding machine ,Pure mathematics ,Mathematics::Dynamical Systems ,Dynamical Systems (math.DS) ,01 natural sciences ,law.invention ,law ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Dynamical Systems ,0101 mathematics ,Spectrum of transition operators ,Mathematics ,Algebra and Number Theory ,Markov chains ,Markov chain ,Mathematics::Operator Algebras ,Fibered Julia sets ,010102 general mathematics ,Spectral properties ,Probabilistic logic ,Julia set ,010307 mathematical physics ,Geometry and Topology ,Stochastic Vershik map ,Bratteli diagrams - Abstract
In this paper, we define some Markov Chains associated to Vershik maps on Bratteli diagrams. We study probabilistic and spectral properties of their transition operators and we prove that the spectra of these operators are connected to Julia sets in higher dimensions. We also study topological properties of these spectra., 34 pages, 19 figures
- Published
- 2021
33. On Julia Limiting Directions in Higher Dimensions
- Author
-
Alastair Fletcher
- Subjects
Polynomial (hyperelastic model) ,Unit sphere ,Quasiconformal mapping ,Sequence ,Mathematics::Dynamical Systems ,Mathematics - Complex Variables ,Mathematics::Complex Variables ,Applied Mathematics ,Image (category theory) ,010102 general mathematics ,Dynamical Systems (math.DS) ,01 natural sciences ,Julia set ,010101 applied mathematics ,Combinatorics ,Computational Theory and Mathematics ,Domain (ring theory) ,FOS: Mathematics ,Mathematics - Dynamical Systems ,Complex Variables (math.CV) ,0101 mathematics ,Analysis ,Meromorphic function ,Mathematics - Abstract
For a quasiregular mapping $$f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}^n$$ , with $$n\ge 2$$ , a Julia limiting direction $$\theta \in S^{n-1}$$ arises from a sequence $$(x_n)_{n=1}^{\infty }$$ contained in the Julia set of f, with $$|x_n| \rightarrow \infty $$ and $$x_n/|x_n| \rightarrow \theta $$ . Julia limiting directions have been extensively studied for entire and meromorphic functions in the plane. In this paper, we focus on Julia limiting directions in higher dimensions. First, we give conditions under which every direction is a Julia limiting direction. Our methods show that if a quasi-Fatou component contains a sectorial domain, then there is a polynomial bound on the growth in the sector. Second, we give a sufficient, but not necessary, condition in $${\mathbb {R}}^3$$ for a set $$E\subset S^2$$ to be the set of Julia limiting directions for a quasiregular mapping. The methods here will require showing that certain sectorial domains in $${\mathbb {R}}^3$$ are ambient quasiballs. This is a contribution to the notoriously hard problem of determining which domains are the image of the unit ball $${\mathbb {B}}^3$$ under an ambient quasiconformal mapping of $${\mathbb {R}}^3$$ onto itself.
- Published
- 2021
34. Fiber Julia sets of polynomial skew products with super-saddle fixed points
- Author
-
Shizuo Nakane
- Subjects
Combinatorics ,Polynomial ,Fiber (mathematics) ,Applied Mathematics ,General Mathematics ,Skew ,Fixed point ,Julia set ,Saddle ,Mathematics - Published
- 2021
35. Orbifold expansion and entire functions with bounded Fatou components
- Author
-
Leticia Pardo-Simón
- Subjects
Pure mathematics ,Class (set theory) ,Mathematics::Dynamical Systems ,Mathematics - Complex Variables ,Applied Mathematics ,General Mathematics ,Entire function ,010102 general mathematics ,Hyperbolic function ,Holomorphic function ,Dynamical Systems (math.DS) ,01 natural sciences ,Julia set ,Bounded function ,0103 physical sciences ,Metric (mathematics) ,FOS: Mathematics ,010307 mathematical physics ,Mathematics - Dynamical Systems ,Complex Variables (math.CV) ,0101 mathematics ,Orbifold ,Mathematics - Abstract
Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are those for which the postsingular set is a compact subset of the Fatou set. Equivalenty, they are characterized as being expanding. Mihaljevi\'c-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalise these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics., Comment: V3: Author accepted manuscript. To appear in Ergod. Theory Dyn. Syst
- Published
- 2021
36. Total disconnectedness of Julia sets of random quadratic polynomials
- Author
-
Krzysztof Lech and Anna Zdunik
- Subjects
Sequence ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dynamical Systems (math.DS) ,Mandelbrot set ,01 natural sciences ,Julia set ,010101 applied mathematics ,Combinatorics ,Cardioid ,Totally disconnected space ,37F35, 37F10 ,FOS: Mathematics ,Almost surely ,Family of sets ,Mathematics - Dynamical Systems ,0101 mathematics ,Dynamical system (definition) ,Mathematics - 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.
- Published
- 2021
37. On uniformly disconnected Julia sets
- Author
-
Vyron Vellis and Alastair Fletcher
- Subjects
Mathematics::Dynamical Systems ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Julia set ,Cantor set ,Set (abstract data type) ,Combinatorics ,Totally disconnected space ,0103 physical sciences ,Uniformization theorem ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes, this result comes down to the fact that such a Julia set is both uniformly perfect and uniformly disconnected. We study the analogous question for Julia sets of UQR maps in $$\mathbb {S}^n$$ , for $$n\ge 2$$ . Introducing hyperbolic UQR maps, we show that the Julia set of such a map is uniformly disconnected if it is totally disconnected. Moreover, we show that if E is a compact, uniformly perfect and uniformly disconnected set in $$\mathbb {S}^n$$ , then it is the Julia set of a hyperbolic UQR map $$f:\mathbb {S}^N \rightarrow \mathbb {S}^N$$ where $$N=n$$ if $$n=2$$ and $$N=n+1$$ otherwise.
- Published
- 2021
38. Global graph of metric entropy on expanding Blaschke products
- Author
-
Yunping Jiang
- Subjects
Pure mathematics ,Applied Mathematics ,Blaschke product ,Quadratic function ,Mandelbrot set ,Julia set ,symbols.namesake ,Unit circle ,symbols ,Discrete Mathematics and Combinatorics ,Gibbs measure ,Real line ,Analysis ,Mathematics ,Analytic function - Abstract
We study the global picture of the metric entropy on the space of expanding Blaschke products. We first construct a smooth path in the space tending to a parabolic Blaschke product. We prove that the metric entropy on this path tends to 0 as the path tends to this parabolic Blaschke product. It turns out that the limiting parabolic Blaschke product on the unit circle is conjugate to the famous Boole map on the real line. Thus we can give a new explanation of Boole's formula discovered more than one hundred and fifty years ago. We modify the first smooth path to get a second smooth path in the space of expanding Blaschke products. The second smooth path tends to a totally degenerate map. We see that the first and second smooth paths have completely different asymptotic behaviors near the boundary of the space of expanding Blaschke products. However, they represent the same smooth path in the space of all smooth conjugacy classes of expanding Blaschke products. We use this to give a complete description of the global graph of the metric entropy on the space of expanding Blaschke products. We prove that the global graph looks like a bell. It is the first result to show a global picture of the metric entropy on a space of hyperbolic dynamical systems. We apply our results to the measure-theoretic entropy of a quadratic polynomial with respect to its Gibbs measure on its Julia set. We prove that the measure-theoretic entropy on the main cardioid of the Mandelbrot set is a real analytic function and asymptotically zero near the boundary.
- Published
- 2021
39. THE SETS OF JULIA AND MANDELBROT FOR MULTI-DIMENSIONAL CASE OF LOGISTIC MAPPING
- Author
-
Rasul Ganikhodzhaev, Shavkat Seytov, Lazizbek Sadullayev, and Islombek Obidjonov
- Subjects
Mathematics::Dynamical Systems ,Logistic mapping ,02 engineering and technology ,General Medicine ,Mandelbrot set ,021001 nanoscience & nanotechnology ,Julia set ,Algebra ,CHAOS (operating system) ,0202 electrical engineering, electronic engineering, information engineering ,Multi dimensional ,Graphical analysis ,020201 artificial intelligence & image processing ,0210 nano-technology ,Mathematics - Abstract
The present paper is devoted to investigation of the multidimensional case of the logistic mapping on the plane to itself. In this paper we learnt the properties of the sets of Julia and Mandelbrot for some two-dimensional logistic mappings. The sets of Julia and Mandelbrot help to define asymptotical behavior of the trajectories of certain mappings. The analytical solutions of the equations for finding fixed and periodic points and the computational simulations for describing the sets of Julia and Mandelbrot are the main results of this paper.
- Published
- 2020
40. Periodic Components of the Fatou Set of Three Transcendental Entire Functions and Their Compositions
- Author
-
Ajaya Singh and Bishnu Hari Subedi
- Subjects
Pure mathematics ,Entire function ,Transcendental number ,Julia set ,Mathematics - Abstract
We prove that there exist three different transcendental entire functions that can have infinite number of domains which lie in the different periodic component of each of these functions and their compositions.
- Published
- 2020
41. Dynamics of generic endomorphisms of Oka-Stein manifolds
- Author
-
Leandro Arosio and Finnur Larusson
- Subjects
Pure mathematics ,Fatou set ,Endomorphism ,Mathematics::Dynamical Systems ,General Mathematics ,Periodic point ,Julia set ,Dynamical Systems (math.DS) ,Domain (mathematical analysis) ,Linear algebraic group ,Stein manifold ,Oka manifold ,FOS: Mathematics ,Complex Variables (math.CV) ,Mathematics - Dynamical Systems ,Mathematics ,Chain-recurrent point ,Mathematics::Complex Variables ,Mathematics - Complex Variables ,Manifold ,Dynamics ,Settore MAT/03 ,Closure (mathematics) ,Bounded function ,Derived set ,Non-wandering point - Abstract
We study the dynamics of a generic endomorphism f of an Oka–Stein manifold X. Such manifolds include all connected linear algebraic groups and, more generally, all Stein homogeneous spaces of complex Lie groups. We give several descriptions of the Fatou set and the Julia set of f. In particular, we show that the Julia set is the derived set of the set of attracting periodic points of f and that it is also the closure of the set of repelling periodic points of f. Among other results, we prove that f is chaotic on the Julia set and that every periodic point of f is hyperbolic. We also give an explicit description of the “Conley decomposition” of X induced by f into chain-recurrence classes and basins of attractors. For $$X={\mathbb {C}}$$ , we prove that every Fatou component is a disc and that every point in the Fatou set is attracted to an attracting cycle or lies in a dynamically bounded wandering domain (whether such domains exist is an open question).
- Published
- 2022
42. Limit drift for complex Feigenbaum mappings
- Author
-
Genadi Levin and Grzegorz Świa̧tek
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Dynamical Systems (math.DS) ,Fixed point ,01 natural sciences ,Measure (mathematics) ,Julia set ,Critical point (thermodynamics) ,0103 physical sciences ,FOS: Mathematics ,Feigenbaum function ,010307 mathematical physics ,Invariant measure ,Mathematics - Dynamical Systems ,0101 mathematics ,Invariant (mathematics) ,Complex plane ,Mathematics - Abstract
We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order \ell of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to 0 under the dynamics of the tower for corresponding \ell. That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when \ell tends to infinity. We also prove the convergence of the drifts to a finite limit which can be expressed purely in terms of the limiting tower which corresponds to a Feigenbaum map with a flat critical point, To appear in Ergodic Theory and Dynamical Systems
- Published
- 2020
43. On the cycles of components of disconnected Julia sets
- Author
-
Wenjuan Peng and Guizhen Cui
- Subjects
Combinatorics ,Connected component ,Mathematics::Dynamical Systems ,Degree (graph theory) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,Construct (python library) ,0101 mathematics ,01 natural sciences ,Julia set ,Mathematics - Abstract
For any integers $$d\ge 3$$ and $$n\ge 1$$ , we construct a hyperbolic rational map of degree d such that it has n cycles of the connected components of its Julia set except single points and Jordan curves.
- Published
- 2020
44. Speiser class Julia sets with dimension near one
- Author
-
Simon Albrecht and Christopher J. Bishop
- Subjects
Class (set theory) ,Mathematics - Complex Variables ,General Mathematics ,Entire function ,010102 general mathematics ,Dynamical Systems (math.DS) ,Primary: 37F10, Secondary: 30D05 ,01 natural sciences ,Julia set ,010101 applied mathematics ,Set (abstract data type) ,Combinatorics ,Singular value ,Dimension (vector space) ,Bounded function ,Hausdorff dimension ,FOS: Mathematics ,Mathematics - Dynamical Systems ,Complex Variables (math.CV) ,0101 mathematics ,Analysis ,Mathematics - Abstract
For any $ \delta >0$ we construct an entire function $f$ with three singular values whose Julia set has Hausdorff dimension at most $1=\delta$. Stallard proved that the dimension must be strictly larger than 1 whenever $f$ has a bounded singular set, but no examples with finite singular set and dimension strictly less than 2 were previously known., Comment: 53 pages, 25 figures
- Published
- 2020
45. A dynamical dimension transference principle for dynamical diophantine approximation
- Author
-
Guohua Zhang and Bao-Wei Wang
- Subjects
Continuous function (set theory) ,General Mathematics ,Diophantine equation ,010102 general mathematics ,Dimension (graph theory) ,Diophantine approximation ,01 natural sciences ,Julia set ,Combinatorics ,Cantor set ,Compact space ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Dynamical system (definition) ,Mathematics - Abstract
Diophantine approximation in dynamical systems concerns the Diophantine properties of the orbits. In classic Diophantine approximation, the powerful mass transference principle established by Beresnevich and Velani provides a general principle to the dimension for a limsup set. In this paper, we aim at finding a general principle for the dimension of the limsup set arising in a general expanding dynamical system. More precisely, let (X, T) be a topological dynamical system where X is a compact metric space and $$T:X\rightarrow X$$ is an expanding continuous transformation. Given $$y_o\in X$$ , we consider the following limsup set $$\mathcal {W}(T,f)$$ , driven by the dynamical system (X, T), $$\begin{aligned} \Big \{x\in X: x\in B(z, e^{-S_n(f+\log |T'|)(z)})\ \text {for some}\ z\in T^{-n}y_o\ {\text {with infinitely many}}\ n\in \mathbb {N}\Big \}, \end{aligned}$$ where $$\log |T'|$$ is a function reflecting the local conformality of the transformation T, f is a non-negative continuous function over X, and $$S_n (f+ \log |T'|) (z)$$ denotes the ergodic sum $$(f+ \log |T'|) (z)+\cdots +(f+ \log |T'|) (T^{n-1} z)$$ . By proposing a dynamical ubiquity property assumed on the system (X, T), we obtain that the dimensions of X and $$\mathcal {W}(T,f)$$ are both related to the Bowen-Manning-McCluskey formulae, namely the solution to the pressure functions $$\begin{aligned} \texttt {P}(-t\log |T'|)=0\ \text {and}\ \texttt {P}(-t(\log |T'|+f))=0, \text {respectively}. \end{aligned}$$ We call this phenomenon a dynamical dimension transference principle, because of its partial analogy with the mass transference principle. This general principle unifies and extends some known results which were considered only separatedly before. These include the b-adic expansions, expanding rational maps over Julia sets, inhomogeneous. Diophantine approximation on the triadic Cantor set and finite conformal iterated function systems.
- Published
- 2020
46. Dynamics on the Pre-periodic Components of the Fatou Set of Three Transcendental Entire Functions and Their Compositions
- Author
-
Bishnu Hari Subedi and Ajaya Singh
- Subjects
Pure mathematics ,Entire function ,Dynamics (mechanics) ,Transcendental number ,Julia set ,Mathematics - Abstract
We prove that there exist three entire transcendental functions that can have an infinite number of domains which lie in the pre-periodic component of the Fatou set each of these functions and their compositions.
- Published
- 2020
47. Criterion for rays landing together
- Author
-
Jinsong Zeng, zeng, jinsong, Laboratoire Angevin de Recherche en Mathématiques (LAREMA), and Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Mathematics::Dynamical Systems ,Applied Mathematics ,General Mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,010102 general mathematics ,[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS] ,Monotonic function ,Dynamical Systems (math.DS) ,[MATH] Mathematics [math] ,Quadratic function ,01 natural sciences ,Julia set ,Combinatorics ,Iterated function ,FOS: Mathematics ,Mathematics - Dynamical Systems ,[MATH]Mathematics [math] ,0101 mathematics ,Complex plane ,Mathematics - Abstract
We give a criterion to determine when two external rays land at the same point for polynomials with locally connected Julia sets. As an application, we provide an elementary proof of the monotonicity of the core entropy along arbitrary veins of the Mandelbrot set., Comment: 22 pages, 2 figures, writing polished, results unchanged. To appear in Transactions of the American Mathematical Society
- Published
- 2020
48. Real Quadratic Julia Sets Can Have Arbitrarily High Complexity
- Author
-
Michael Yampolsky and Cristobal Rojas
- Subjects
FOS: Computer and information sciences ,Discrete mathematics ,Class (set theory) ,Computational complexity theory ,Applied Mathematics ,Numerical analysis ,Quadratic map ,Dynamical Systems (math.DS) ,010103 numerical & computational mathematics ,Computational Complexity (cs.CC) ,68Q17 and 37E05 ,16. Peace & justice ,01 natural sciences ,Julia set ,Renormalization ,Computer Science - Computational Complexity ,Computational Mathematics ,Quadratic equation ,Computational Theory and Mathematics ,High complexity ,FOS: Mathematics ,Mathematics - Dynamical Systems ,0101 mathematics ,Analysis ,Mathematics - Abstract
We show that there exist real parameters $c$ for which the Julia set $J_c$ of the quadratic map $z^2+c$ has arbitrarily high computational complexity. More precisely, we show that for any given complexity threshold $T(n)$, there exist a real parameter $c$ such that the computational complexity of computing $J_c$ with $n$ bits of precision is higher than $T(n)$. This is the first known class of real parameters with a non poly-time computable Julia set., 9 pages, 1 figure. To be published in the journal Found. Comp. Math. (FoCM). arXiv admin note: text overlap with arXiv:1703.04660
- Published
- 2020
49. Wiman–Valiron Discs and the Dimension of Julia Sets
- Author
-
Waterman, James
- Subjects
Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,01 natural sciences ,Julia set ,010101 applied mathematics ,Combinatorics ,Singular value ,Bounded function ,Hausdorff dimension ,Simply connected space ,0101 mathematics ,Orbit (control theory) ,Mathematics ,Meromorphic function - Abstract
We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman–Valiron theory.
- Published
- 2020
50. Mandelbrot Sets and Julia Sets in Picard-Mann Orbit
- Author
-
Abdul Aziz Shahid, Arshad Khan, Maqbool Ahmad, Asifa Tassaddiq, and Cui Zou
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,General Computer Science ,General Engineering ,Julia set ,020207 software engineering ,02 engineering and technology ,Mandelbrot set ,escape criterion ,01 natural sciences ,010305 fluids & plasmas ,Fractal ,Computer Science::Graphics ,0103 physical sciences ,0202 electrical engineering, electronic engineering, information engineering ,General Materials Science ,lcsh:Electrical engineering. Electronics. Nuclear engineering ,Orbit (control theory) ,Picard-Mann iteration ,lcsh:TK1-9971 ,Mathematics - Abstract
The purpose of this paper is to introduce the Mandelbrot and Julia sets by using Picard-Mann iteration procedure. Escape criteria is established which plays an important role to generate Mandelbrot and Julia sets. Also, numerous graphical pictures of these sets have been visualized and certain examples have been recognized. Presented results shows that fractal images generated by Picard-Mann iteration procedure are entirely different from those generated in Mann orbit.
- Published
- 2020
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