ShellThron
Idea
In complex analysis the exponential map is studied where and . This gives up power towers of to height , or .
This provides us with the tetration Mandelbrot escape fractal in Figures 1 and tetration Mandelbrot period fractal in Figure 2.
The tetration Mandelbrot escape fractal in Figure 1 is the escape value of where . When 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 ; 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 .
Formal Definition
The red area of Figure 2 displays the values for which is period 1, in other words convergent. This is called the ShellThron region.
As in complex dynamics, let with . Let , then is a fixed point of . The dynamics of the immediate region around a fixed point is where is called the Lyapunov multiplier . When the point is an attractor and when the point is a repellor. The exceptional points are where , thus for and .
Let be a complex function with , such that . At the fixed point we have . 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
 Hyperbolic attractor
 Hyperbolic repellor
 Parabolic Neutral
 Rationally Neutral where
 Irrationally Neutral and not rationally neutral
Derivation of ShellThron boundary
Now let , where . Then and at and . So we are interested in where or .
Since , we have as the boundary of the ShellThron region.

Examples
Properties
Related Entries
References
Shell, Donald L. “On the convergence of infinite exponentials.” Proceedings of the American Math. Society 13 (1962), pp. 678681.
Thron, W. J. “Convergence of infinite exponentials with complex elements.” Proceedings of the American Math. Society 8 (1957), pp. 10401043.