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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: They seem to reference: But what is a ...

Let $g(x) = x^6 - 30 x $ Let $h(x) = x^6 $ Let $f(x) = x^2 - 2 $ Let $r$ be a reduced fraction $0 < \frac{p}{q} < 2 $ with integers $p,q > 1$ Let $f_{n+1}(x) = f(f_n(x)) = f_n(f(x)) , ...

What are the bounds on the possible values of the Hausdorff dimension of the Julia sets of quadratics not in the Mandelbrot set? In particular, assume we have a quadratic $q_c: z \mapsto z^2 + c$ on ...