The Mandelbrot Set and Julia Sets

Julia Sets - Computational Issues

Run away to infinity

How does a sequence run away to infinity anyway?

Must we check a very, Very, VERY, VERY long time?

Here some mathematical reasoning helps with the interpretation. It is not difficult to prove that if some member z_j of the sequence is farther than 2 from the origin, then the distance between the origin and later members of the sequence will grow without bound.

This is what we mean by run away to infinity, and so all we have to check is whether the sequence runs farther away than 2.

Here are some iterates of z_n+1 = z_n² + c for c = -0.25 + 0.25i, starting with z₀ = 0.5 + 0.7i. The points z₀ through z₄ are shown, with later iterates a brighter red.

Note that z₄ is outside the circle of radius 2, so later z_i should run farther away from the origin. A few more iterates will illustrate this.

point	distance to the origin
z0 = 0.5 + 0.7i	0.8602
z1 = -0.49 + 0.95i	1.0689
z2 = -0.9124 - 0.681i	1.1385
z3 = 0.1187 + 1.493i	1.4974
z4 = -2.4640 + 0.6066i	2.5371
z5 = 5.4561 - 2.7285i	6.1003
z6 = 22.0745 - 29.5244i	36.8642
z7 = -384.658 - 1303.22i	1358.8
z8 = -1.5504*106 + 1.0026*106	1.8463*106

Consequently we call this condition

some z_j is farther than 2 from the origin

the escape criterion.

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