0

I have met five different definitions of the Julia set and I am trying to work out why they are equivalent. I haven't managed to find a reference showing why two of these are equivalent.

Why is the boundary of the basin of attraction of $\infty$ equal to the closure of the set of repelling periodic points?

I see the connection since if $|f'(z_0)|<1$ then there is a neighbourhood of $z_0$ all of whose iterates will remain in the neighbourhood so the attracting fixed points are not even near to diverge to infinity. If $|f'(z_0)|>1$ then there is neighbourhood all of whose points except $z_0$ will "converge" to $\infty$. So the "except" case would be the boundary case?

user30523
  • 1,681
  • I don't think there's any short answer to this question. I'd have a look at the text Complex Dynamics by Carleson and Gamelin. This result appears in the first paragraph of chapter III section 4 on page 65 of that text but it really does build on quite a lot of the results prior to that. It's a corollary to a similar result for rational functions and I don't think there's any major simplification made possible to restricting to polynomials. – Mark McClure Feb 04 '19 at 10:30
  • Great, the reference was very helpful. I had managed to miss that result. – user30523 Feb 12 '19 at 20:38

0 Answers0