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What are some examples of complex polynomials whose Julia sets are connected, but not locally?

In the book Complex Dynamics by Carleson and Gamelin, I found: enter image description here

They seem to reference: enter image description here

But what is a specific value of $\lambda$ that satisfies the hypotheses?

Ultimately I want to generate pictures of non-locally connected Julia sets, so I'm interested in any polynomials (in addition to $\lambda z+z^2$) with this property.

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There is a necessary and sufficient condition due to Yoccoz in terms of continued fraction expansion. The condition is that $\lambda=e^{2\pi i \theta}$ should be such that $\theta$ is not a Brjuno number, see

https://en.wikipedia.org/wiki/Brjuno_number

for more details and examples.

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  • $\begingroup$ Did you mean such that $\vartheta$ is not a Brjuno number? $\endgroup$
    – D.S. Lipham
    Commented Oct 11 at 21:42
  • $\begingroup$ @D.S.Lipham yes, corrected, thanks! $\endgroup$
    – Malik Younsi
    Commented Oct 12 at 7:52
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(More of a comment, but making this an answer to include an image.)

One nice reference is these slides from Milnor, which include theorems about non-locally-connected Julia sets, and some specific visualizations based on an iterated process. Here's an image from the slides that gives an intuition for what such Julia sets look like.

As Milnor points out, any visualization is just going to be an approximation. But if you examine the spirals, it makes sense this isn't a locally connected set in the limit.

Non-locally-connected Julia set, looks like many very tight spirals

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    $\begingroup$ Milnor's slide is about non-locally connected Julia sets for infinitely renormalizable polynomials, not for irrationally indifferent fixed point. Infinite renormalization is another source of complicated interesting phenomena, and arguments are often similar for both cases. $\endgroup$
    – inyo
    Commented Oct 22 at 2:10

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