Abstract
Centered polygonal lacunary functions are a type of lacunary function that exhibit behaviors that set them apart from other lacunary functions, this includes rotational symmetry. This work will build off of earlier studies to incorporate the automorphism group of the open unit disk D, which is a subgroup of the Möbius transformations. The behavior, dimension, dynamics, and sensitivity of filled-in Julia sets and Mandelbrot sets to variables will be discussed in detail. Additionally, several visualizations of this three-dimensional parameter space will be presented.
Highlights
The example above has a natural boundary at the unit circle and is analytic on the open unit disk
The lacunary Möbius fractals studied in the current work show global similarity between the Mandelbrot and Julia sets, albeit for different reasons
In order to maintain a limit in scope, this work will exclusively focus on the the fractal character of the filled-in Julia sets and their corresponding Mandelbrot sets [21,22,23]
Summary
In a previous work the current authors explored some of the fractal character of the centered polygonal lacunary functions [2]. In that work both the “classical” Julia set, generated via direct iteration of the function, and a “phase-rotated” Julia set were explored. The lacunary Möbius fractals studied in the current work show global similarity between the Mandelbrot and Julia sets, albeit for different reasons. | f1(6k)(z)| for k = 1, k = 2, and k = 3 are shown in the right-hand column of Figure 1 These contour plots expose an important property of centered polygonal based lacunary function.
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