Exploration of Filled-In Julia Sets Arising from Centered Polygonal Lacunary Functions

Mork, L.K.; Vogt, Trenton; Sullivan, Keith; Rutherford, Drew; Ulness, Darin J.

doi:10.3390/fractalfract3030042

Cited by 8 publications

(22 citation statements)

References 30 publications

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“…The inherent organization of the centered polygonal lacunary sequences gives rise to fractal-like self-similarity scaling features. Indeed, the current authors have recently investigated fractal and scaling behavior of centered polygonal lacunary functions [26,27]. It is hoped that the current work will add to the foundational understanding of that earlier work.…”

Section: Introductionmentioning

confidence: 82%

Centered Polygonal Lacunary Sequences

2019

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Lacunary functions based on centered polygonal numbers have interesting features which are distinct from general lacunary functions. These features include rotational symmetry of the modulus of the functions and a notion of polished level sets. The behavior and characteristics of the natural boundary for centered polygonal lacunary sequences are discussed. These systems are complicated but, nonetheless, well organized because of their inherent rotational symmetry. This is particularly apparent at the so-called symmetry angles at which the values of the sequence at the natural boundary follow a relatively simple 4 p -cycle. This work examines special limit sequences at the natural boundary of centered polygonal lacunary sequences. These sequences arise by considering the sequence of values along integer fractions of the symmetry angle for centered polygonal lacunary functions. These sequences are referred to here as p-sequences. Several properties of the p-sequences are explored to give insight in the centered polygonal lacunary functions. Fibered spaces can organize these cycles into equivalence classes. This then provides a natural way to approach the infinite sum of the actual lacunary function. It is also seen that the inherent organization of the centered polygonal lacunary sequences gives rise to fractal-like self-similarity scaling features. These features scale in simple ways.

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Section: Introductionmentioning

confidence: 82%

Centered Polygonal Lacunary Sequences

2019

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“…The exploration of filled-in Julia sets arising from centered polygonal lacunary functions find in [47]. It is determined in [7] that Newton maps of rational functions are conjugate to quadratic polynomials.…”

Section: Different Approaches To Studying Dynamics Of One Variable Co...mentioning

confidence: 99%

On Some Advances in Dynamics of One Variable Complex Functions

Sajid¹

2022

Int. J. of Appl. Math.

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In this article, we present some recent advances in the dynamics of one variable complex functions which are especially associated to the Julia sets, the Fatou sets, the Sigel disks, the Herman rings, fixed points as well as the Mandelbrot sets. For the sake of interest, we consider mainly research works which have been done from 2015 to 2019. To achieve our purpose, some interesting and selected themes are considered to show some recent advances in the dynamics of one variable complex functions.

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“…Thus, the act of raising the power on the prime is akin to zooming in on a fractal object. Relatedly, the actual fractal character as manifested by Julia sets for the centered polygonal lacunary functions has recently been investigated [26].…”

Section: Sprays Renormalization and Fractal Charactermentioning

confidence: 99%

Centered Polygonal Lacunary Graphs: A Graph Theoretic Approach to p-Sequences of Centered Polygonal Lacunary Functions

2019

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This work is on the nature and properties of graphs which arise in the study of centered polygonal lacunary functions. Such graphs carry both graph-theoretic properties and properties related to the so-called p-sequences found in the study of centered polygonal lacunary functions. p-sequences are special bounded, cyclic sequences that occur at the natural boundary of centered polygonal lacunary functions at integer fractions of the primary symmetry angle. Here, these graphs are studied for their inherent properties. A ground-up set of planar graph construction schemes can be used to build the numerical values in p-sequences. Further, an associated three-dimensional graph is developed to provide a complementary viewpoint of the p-sequences. Polynomials can be assigned to these graphs, which characterize several important features. A natural reduction of the graphs original to the study of centered polygonal lacunary functions are called antipodal condensed graphs. This type of graph provides much additional insight into p-sequences, especially in regard to the special role of primes. The new concept of sprays is introduced, which enables a clear view of the scaling properties of the underling centered polygonal lacunary systems that the graphs represent. Two complementary scaling schemes are discussed.

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Exploration of Filled-In Julia Sets Arising from Centered Polygonal Lacunary Functions

Cited by 8 publications

References 30 publications

Centered Polygonal Lacunary Sequences

Centered Polygonal Lacunary Sequences

On Some Advances in Dynamics of One Variable Complex Functions

Centered Polygonal Lacunary Graphs: A Graph Theoretic Approach to p-Sequences of Centered Polygonal Lacunary Functions

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