# Shell-Thron

## Idea

In complex analysis the exponential map is studied where ${\displaystyle f(z)=a^{z}}$ and ${\displaystyle a,z\in \mathbb {C} }$. This gives up power towers of ${\displaystyle a}$ to height ${\displaystyle n}$, or ${\displaystyle ^{n}a=a^{{\Bigl (}{a^{\cdot ^{\cdot ^{a}}}}{\Bigr )}}}$.

This provides us with the tetration Mandelbrot escape fractal in Figures 1 and tetration Mandelbrot period fractal in Figure 2.

 Figure 1: Tetration Mandelbrot escape fractal Figure 2: Tetration Mandelbrot period fractal

The tetration Mandelbrot escape fractal in Figure 1 is the escape value of ${\displaystyle n}$ where ${\displaystyle |^{n}a|\geq 10^{6}}$. When ${\displaystyle ^{n}a}$ is periodic it doesn't escape. This leads to a complementary fractal, the tetration Mandelbrot period fractal in Figure 2. This fractal displays the period of ${\displaystyle ^{n}a}$; moving through the colors of the rainbow, red is period 1, yellow is period 2, green is period 3 and so on.

When the values of while the tetration Julia fractal displays the dynamics of the map ${\displaystyle z\to (-1)^{z}}$.

## Formal Definition

The red area of Figure 2 displays the values for which ${\displaystyle ^{n}a}$ is period 1, in other words convergent. This is called the Shell-Thron region.

As in complex dynamics, let ${\displaystyle z,z_{0}\in \mathbb {C} }$ with ${\displaystyle f:\mathbb {C} \to \mathbb {C} }$. Let ${\displaystyle f(z_{0})=z_{0}}$, then ${\displaystyle z_{0}}$ is a fixed point of ${\displaystyle f}$. The dynamics of the immediate region around a fixed point ${\displaystyle z_{0}}$ is ${\displaystyle f^{n}(z-z_{0})=z_{0}+f'(z_{0})^{n}(z-z)+O((z-z_{0})^{2})}$ where ${\displaystyle f'(z_{0})}$ is called the Lyapunov multiplier ${\displaystyle \lambda }$. When ${\displaystyle |\lambda |<1}$ the point ${\displaystyle z_{0}}$ is an attractor and when ${\displaystyle |\lambda |>1}$ the point ${\displaystyle z_{0}}$ is a repellor. The exceptional points are where ${\displaystyle |\lambda |=1}$, thus ${\displaystyle \lambda =e^{2\pi ix}}$for ${\displaystyle x\in \mathbb {R} }$ and ${\displaystyle 0\leq x\leq 1}$.

Let ${\displaystyle f:\mathbb {C} \to \mathbb {C} }$ be a complex function with ${\displaystyle L\in \mathbb {C} }$, such that ${\displaystyle f(L)=L}$. At the fixed point we have ${\displaystyle f^{n}(L)=L}$. The most important principle in fractional iteration and extending tetration is that the dynamics of iterated functions simplify and become "linear" in the neighborhood of a fixed point.

### Classification of Complex Fixed Points

• Superattracting ${\displaystyle f'(z_{0})=0}$
• Hyperbolic attractor ${\displaystyle |f'(z_{0})|<1}$
• Hyperbolic repellor ${\displaystyle |f'(z_{0})|>1}$
• Parabolic Neutral ${\displaystyle f'(z_{0})=1}$
• Rationally Neutral ${\displaystyle f'(z_{0})^{k}=1}$ where ${\displaystyle k\in \mathbb {Z} ^{+}}$
• Irrationally Neutral ${\displaystyle |f'(z_{0})|=1}$ and not rationally neutral

### Derivation of Shell-Thron boundary

Now let ${\displaystyle f(z)=a^{z}}$, where ${\displaystyle a\in \mathbb {C} }$. Then ${\displaystyle {\frac {d}{dz}}a^{z-z_{0}}={\frac {d}{dz}}e^{\log(a)(z-z_{0})}=\log(a)a^{z-z_{0}}}$ and at ${\displaystyle z=z_{0}}$ and ${\displaystyle f'(z_{0})=\log(a)z_{0}=\log(a^{z_{0}})=\log(z_{0})}$. So we are interested in where ${\displaystyle \log(z_{0})=e^{2\pi ix}}$ or ${\displaystyle z_{0}=e^{(e^{2\pi ix})}}$.

Since ${\displaystyle a^{z_{0}}=z_{0}}$, we have ${\displaystyle a={z_{0}}^{\frac {1}{z_{0}}}=e^{(e^{2\pi ix-(e^{2\pi ix})})}}$ as the boundary of the Shell-Thron region.

Figure 4: Shell-Thron boundary
 Tetration with the Shell-Thron region in red

## References

Shell, Donald L. “On the convergence of infinite exponentials.” Proceedings of the American Math. Society 13 (1962), pp. 678-681.

Thron, W. J. “Convergence of infinite exponentials with complex elements.” Proceedings of the American Math. Society 8 (1957), pp. 1040-1043.