The Mandelbrot Set and Julia Sets

"Run Away to Infinity" Criterion

Here we show that if some z_n is farther than 2 from the origin, then successive iterates will grow without bound. That is, they will run away to ∞.

For a complex number z_n = x_n + i⋅y_n, the absolute value is

|z_n| = √(x_n² + y_n²),

the distance from z_n to the origin.

Recalling the sequence z₀, z₁, ... is defined by z_n+1 = z_n² + c, we show if some z_n satisfies |z_n| > max(2, |c|), then the sequence z_n, z_n+1, ... runs away to ∞.

So suppose |z_n| > max(2, |c|).

Because |z_n| > 2, we can write

|z_n| = 2 + ε,

for some ε > 0.

Now

|z_n²| = |z_n² + c - c| ≤ |z_n² + c| + |c|

So

|z_n² + c| ≥ |z_n²| - |c| = |z_n|² - |c|

> |z_n|² - |z_n| (because |z_n| > |c|)

= (|z_n| - 1)⋅|z_n| = (1 + ε)⋅|z_n|

That is, |z_n+1| > (1 + ε)⋅|z_n|. Iterating, |z_n+k| > (1 + ε)^k⋅|z_n|.

To complete the proof that |z_n| > 2 implies the sequence runs away to infinity, observe that if |c| > 2, then

z₀ = 0

z₁ = c

and z₂ = c² + c = c⋅(c + 1)

so |z₂| = |c|⋅|c + 1| > |c| (noting |c + 1| > 1 because |c| > 2).

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