Projective Fractals

z (z1/z) (Log(z1/z)) (Log2(z1/z))

The first row displays periods and the second row displays the complementary escaping fractals. The first column is the standard tetration fractal. The area of convergence in red is bounded by Equation 1 where 0<= x <= 1 and displayed in Figure 1. The points at x = 1/n are on the boundary between period 1 and period n.

Equation 1

Figure 1

The second column is a projection of the first such that the inner red area displays the location of the fixed points of the first set of fractals. The boundary for the area of fixed points for the area of convergence is given by Equation 2 where 0<= x <= 1 and displayed in Figure 2
Equation 2

Figure 2

The third column is based on the Lyapunov characteristic number where the bounding curve is the unit circle. The fourth column is based on the Lyapunov exponent and forms a line on the axis of pure imaginary numbers. This is a nice demonstration of the area of convergence of the tower function. It is also a nice demonstration of the limitation of visual "proofs" since the point at -1 is convergent but its convergence is not visually apparent.