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.
Figure 1: Tetration Mandelbrot escape fractal
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Figure 2: Tetration Mandelbrot period fractal
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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 Shell-Thron 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 Shell-Thron boundary
Now let , where . Then and at and . So we are interested in where or .
Since , we have as the boundary of the Shell-Thron region.
Figure 4: Shell-Thron boundary
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Tetration with the Shell-Thron region in red
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Examples
Properties
Related Entries
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.