The Escape Criterion

There is a useful criterion for the escape of a complex number under the quadratic family. It is embodied in the following theorem:

Theorem:If tex2html_wrap_inline258 then tex2html_wrap_inline260 as tex2 html_wrap_inline262

Proof:

Now, by the Triangle Inequality, tex2html_wrap_inline264 So tex2html_wrap_inline266 Since tex2html_wrap_inline268 there exists a tex2html_wrap_inline270 such that tex2html_wr ap_inline272 So tex2html_wrap_inline274 So tex2html_wrap_inline276 We may repeat this argument repeatedly to find that tex2html_wrap_inline278 for all n. Since tex2html_wrap_inline282

Q.E.D.

There is a corollary which will prove important later in this paper.

Corollary:If tex2html_wrap_inline284 then the orbit of 0 escapes to infinity under tex2html_wrap_inline286

Proof:

tex2html_wrap_inline288 so by the theorem, the orbit of 0 escapes to infinity under tex2html_wrap_inline152

Q.E.D.

A corollary which follows from this theorem which will be stated without proof is

Corollary:If for some tex2html_wrap_inline292 then tex2html_wrap_inline260 as tex2html_wrap_inline262

The above corollary leads easily to an algorithm to draw the filled Julia Set of tex2html_wrap_inline152 We simply pick a grid of points and iterate each point, say, 20 times. If the 20th iterate is outside the circle of radius 2, then the orbit escapes. We may use different colors to represent the number of iterations it took to escape. Notice, however, that this is not foolproof. It may take more that 20 iterations for us to get an accurate picture of the dynamics. However, 20 iterations is a good approximation. The drawings which follow use this algorithm.