# Lacunary Möbius Fractals on the Unit Disk

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## Abstract

**:**

## 1. Introduction

## 2. Centered Polygonal Lacunary Functions

#### Jacobi $\vartheta $-Functions

## 3. Iteration of Centered Polygonal Lacunary Functions

## 4. Julia Sets and Mandelbrot Sets

#### 4.1. Three-Dimensional and Two-Dimensional Mandelbrot Sets

#### 4.2. Filled-In Julia Sets

#### 4.3. Dynamicism of the Filled-In Julia Sets

#### 4.4. Fractal Dimension

## 5. Code Snippets

`JSplotallMob[k_, m_, j_, s_, p1_, p2_] :=`

`Module[{f},`

`f[y_] := GF[p1 (y - p2)/(Conjugate[p2] y - 1), (k n^2 - k n + 2)/2,`

`1, m]; ListDensityPlot[`

`Flatten[Table[`

`Table[{x, y, If[Abs[Nest[f, z, j]] > 2, 1, 0]}, {x, -1.1, 1.1,`

`s}], {y, -1.1, 1.1, s}], 1], ColorFunction -> GrayLevel] //`

`Quiet]`

`GF[z_, g_, p1_, p2_] := Sum[z^g, {n, p1, p2}]`

`Madelbrot3D =`

`Flatten[Table[`

`Table[Table[{x, y, j,`

`If[x^2 + y^2 > 1, 1,`

`ManMobpt[1, 14, 25, Exp[I j Pi], x + I y ]]}, {x, -1,`

`1, .025}], {y, -1, 1, .025}], {j, 0, 2, .025}], 2]`

`ManMobpt[k_, m_, j_, p1_, p2_] :=`

`Module[{f},`

`f[y_] := GF[p1 (y - p2)/(Conjugate[p2] y - 1), (k n^2 - k n + 2)/2,`

`1, m]; If[Abs[Nest[f, 0, j]] > 2, 1, 0] // Quiet]`

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**One nice way to present graphs of centered polygonal functions is shown on the right side column. The representation given here have been described in previous work [1,2]. The contour plot of the modulus is truncated at the unity level set (blue shading represents low values and red shading represents high values). The left side shows the example in which $k=4$ where a surface plot of $|{f}_{16}\left(z\right)|$ is shown above the contour plot as a guide to the interpretation of the contour plot. The unit circle is superimposed in black to guide the eye. The right column of panels show $|{f}_{16}\left(z\right)|$ for the cases of $k=1$, $k=2$, and $k=3$. The k-fold rotational symmetry of the the centered polygonal lacunary functions are clearly evident.

**Figure 2.**The “surface-swarm” (top row and bottom right) and “cylinder” (bottom left) graphical representation of ${\mathfrak{M}}_{4}$ for ${}^{25}{h}_{14}^{\left(4\right)}$. The sampling step size is 0.025 in each dimension. For the surface-swarm plots, the Mandelbrot set is colored based how many points in a small region are not in the same set, ${\mathfrak{M}}_{4}$ or its complement. The green (blue) gives the interior (exterior) side of the surface of ${\mathfrak{M}}_{4}$. (The red is an artifact of plotting and not physically relevant). For the cylinder graph, ${\mathfrak{M}}_{4}$ is colored dark grey and its complement is colored light grey. In all cases the plotted regions is $[0,2\pi )\times \mathbb{D}$. Note: the $\theta $-axis is in “units” of $\pi $.

**Figure 3.**Examples of typical and useful two-dimensional subsets of ${\mathfrak{M}}_{4}$. The subsets arise from planar cuts in ${\mathfrak{M}}_{4}$. The resulting two-dimensional Mandelbrot sets can be explored in more detail and at much higher resolution to provide additional insight into lacunary Möbius fractals. Some of these subests are shown in Figure 4 and Figure 5. Members of the two-dimensional Mandelbrot set are colored black and members of the complement are colored white.

**Figure 4.**Two dimensional subsets of ${\mathfrak{M}}_{k}$ for $k=2$ (top row), $k=3$ (second row), $k=4$ (third row), and $k=6$ (bottom row). In the left most column $\theta =0$, $\theta =\frac{\pi}{3}$ in the second column, $\theta =\frac{2\pi}{3}$ in the third column, and $\theta =\pi $ in the right most column. The phase breaks the dihedral symmetry but maintains the k-fold rotational symmetry introduces. Other features of these two Mandelbrot sets are described in the text. Members of the two-dimensional Mandelbrot set are colored black and members of the complement are colored white. In all cases, ${}^{25}{h}_{14}^{\left(k\right)}$ with a sampling step size of 0.005 in both dimensions.

**Figure 5.**Two-dimensional subsets of ${\mathfrak{M}}_{k}$ for $k=2$ (leftmost column), $k=3$ (second column), $k=4$ (third column), and $k=6$ (rightmost column). The top row has $\theta $ as the ordinate (scaled by $\pi $) and Re$\left[a\right]$ as the abscissa and Im$\left[a\right]=0$. The second row has $\theta $ as the ordinate and Im$\left[a\right]$ as the abscissa and Re$\left[a\right]=0$. The features are these two-dimensional Mandelbrot sets are discussed in the texts. Members of the two-dimensional Mandelbrot set are colored black and members of the complement are colored white. In all cases, ${}^{25}{h}_{14}^{\left(k\right)}$ with a sampling step size of 0.005 in both dimensions.

**Figure 6.**Two-dimensional subsets of ${\mathfrak{M}}_{2}$ that will serve as the primary exploration landscape for analysis of the associated filled in Julia sets. The red line indicates the path through parameter space, $\gamma =(1.35\pi ,0.315+iy)$. The green line indicates the path $\gamma =(\theta ,0.315+i0)$. The left panel shows a $\theta $-y plane slice of ${\mathfrak{M}}_{2}$ and the middle panel shows a x-y plane slice. The cyan-colored box in the middle graph indicate a blow-up window which is shown in the right panel. Note, in the graphs, $\theta $ is scaled by $\pi $. The jet structure emanating from the nexus of the main lobes of the Mandelbrot set is prominently featured. Members of the two-dimensional Mandelbrot set are colored black and members of the complement are colored white.

**Figure 7.**Filled-in Julia sets corresponding to equally spaced points in ${\mathfrak{M}}_{2}$ over the range shown in the inset of Figure 8 (0 to $0.08$) along the red path shown in Figure 6. The global form of filled-in Julia sets can change abruptly along this path. In all cases, ${}^{25}{h}_{14}^{\left(2\right)}$ with a sampling step size of 0.005 in both dimensions.

**Figure 8.**The effect of N (

**left**) and j (

**right**) on the JS-derivative. The abscissa for both graphs is y and the ordinate are in arbitrary units. The sampling step size is 0.001. Shown is ${\mathsf{\Delta}}_{2}(1.35\pi ,0.315+iy)$ (the red line in Figure 6). The left graph shows curves for $j=25$ and $N=8$ (brown) and $N=20$ (purple). The JS derivative is quite robust with changing values of N. The right graph shows curves for $N=14$ and $j=15$ (purple), $j=25$ (magenta), $j=35$ (brown), and $j=45$ (blue). The JS-derivatives are quite sensitive to changing values of j. Notably, some peaks have strong j dependence while others have very little j dependence. The inset in the right graph shows a higher resolution scan (step size 0.00025) of the region from $y=0.01$ to $0.08$ using higher values of j; $N=14$ and $j=45$ (blue), $j=60$ (green), $j=75$ (red). The higher resolution reveals structure in the sharp leftmost peak. Note: this peak is referred to as peak 1 in the body of the text.

**Figure 9.**Derivative and dimension data for the green path in Figure 6 which runs through the wispy fray area of ${\mathfrak{M}}_{2}$. The sampling step size is 0.002 for both graphs. The abscissa for both graphs is $\theta $ (scaled by $\pi $) and the ordinate for the left graph is in arbitrary units while that for the right graph is fractal dimension. The second row is a sequence of filled-in Julia sets correspond to the green points (read left-to-right) in the JS-derivative graph ($\theta =0$, $0.1\pi $, $0.2\pi $ and $0.3\pi $). The third row is a sequence corresponding to the magenta points ($\theta =0.382\pi $, $0.428\pi $, $0.460\pi $ and $0.506\pi $).

**Figure 10.**Dimension data for the same path through ${\mathfrak{M}}_{2}$ and parameters as in the right graph in Figure 8 ($N=14$ and $j=15$ (purple), $j=25$ (magenta), $j=35$ (brown), and $j=45$ (blue)). The dependence on j is evident. The right graph shows the dimension of ${}^{25}{h}_{14}^{\left(2\right)}$ over the range $0.01\le y\le 0.07$ (solid black curve). The dotted curve is a that of the JS-derivative (inset of Figure 8 and normalised to fit on the graph). Abrupt changes in JS-derivative appear to be correlated with abrupt changes in dimension.

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**MDPI and ACS Style**

Mork, L.K.; Sullivan, K.; Ulness, D.J.
Lacunary Möbius Fractals on the Unit Disk. *Symmetry* **2021**, *13*, 91.
https://doi.org/10.3390/sym13010091

**AMA Style**

Mork LK, Sullivan K, Ulness DJ.
Lacunary Möbius Fractals on the Unit Disk. *Symmetry*. 2021; 13(1):91.
https://doi.org/10.3390/sym13010091

**Chicago/Turabian Style**

Mork, L. K., Keith Sullivan, and Darin J. Ulness.
2021. "Lacunary Möbius Fractals on the Unit Disk" *Symmetry* 13, no. 1: 91.
https://doi.org/10.3390/sym13010091