Julia set connectivity

test-prep

#1
  • Show that the Julia set of z^2-i is connected.
  • Show that the Julia set of z^2+1 is totally disconnected.

#2

By Theorem 3.5.1, it is sufficient to show that the orbit of the critical zero remains bounded by the escape radius under the iteration of f_{-i}(z) =z^2-i.

For this problem, R = \text{max}(2,|-i|) implies R = 2.

For f_{-i}(z) = z^2 -i, our critical point is 0.

Iterating from 0 gives the sequence \{0, -i, -(1+i), i ,-(1+i), i,...\}. So we are eventually periodic and our orbit is bounded by R. Thus, f_{-i} is connected.

Similarly, by Theorem 3.5.1, it is sufficient to show that some iterate of f_1(z) = z^2 + 1 exceeds the escape radius, R = 2.

Iterating from the critical point, 0, gives the sequence \{0, 1, 2, 5,...\}. So there is an iterate in our orbit that eventually exceeds R. Thus, f_1 is a Cantor Set, or totally disconnected.