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 zj 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 zn+1 = zn2 + 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 zi 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⋅106i 1.8463*106

Consequently we call this condition

some zj is farther than 2 from the origin

the escape criterion.

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