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

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

doi:10.3390/fractalfract3030042

Abstract

Centered polygonal lacunary functions are a particular type of lacunary function that exhibit properties which set them apart from other lacunary functions. Primarily, centered polygonal lacunary functions have true rotational symmetry. This rotational symmetry is visually seen in the corresponding Julia and Mandelbrot sets. The features and characteristics of these related Julia and Mandelbrot sets are discussed and the parameter space, made with a phase rotation and offset shift, is intricately explored. Also studied in this work is the iterative dynamical map, its characteristics and its fixed points.

Highlights

Analytic functions have long played an important role in physics
These isolated singularities restrict the radius of convergence of the power series associated with the analytic functions
The main result of this work is the detailed look at the fractal character of the centered polygonal lacunary functions

Summary

Introduction

Isolated singularities of analytic functions often provide physical insight because they carry much of the information about the function itself These isolated singularities restrict the radius of convergence of the power series associated with the analytic functions. Lacunary functions are a important class of functions that exhibit a natural boundary [1,3,4]. This work is focused on the special family of lacunary functions that occur when the powers are given by the centered polygonal numbers. These are the monotonically increasing sequence of numbers associated with the points on a polygonal lattice [5,6]. The family is referred to as the centered polygonal lacunary functions

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Conclusion