Shell-Thron

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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
Figure 2: Tetration Mandelbrot period fractal

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
Tetration with the Shell-Thron region in red

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.