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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 the boundary of the Mandelbrot set $M$ and a curve $\gamma$ parameterized by the closed unit interval and with $\gamma(t)$ in $\mathbb{C} \setminus M$ for $t< 1$ and $\gamma(1) = c$. If the Hausdorff dimension of $q_c$ is $h$, can one assume that the Hausdorff dimension of the Julia set of $q_t: z \mapsto z^2 + \gamma(t)$ tends to $h$ as $t$ tends to $1$?

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2 Answers

up vote 3 down vote accepted

The Hausdorff dimension of Julia sets of quadratic polynomials has been well-studied, although some questions still remain.

You specifically asked about parameters $c$ that do not belong to the Mandelbrot set. In this case, the map $q_c(z) = z^2 + c$ has a totally disconnected Julia set. Here is what can be said.

1) The Hausdorff dimension is always strictly greater than zero. (This is true for all non-linear, non-constant rational functions, even for meromorphic functions, as proved by Stallard. See e.g. Corollary 2.11 in my paper "Hyperbolic dimension and radial Julia sets of transcendental functions", Proc. Amer. Math. Soc. 137 (2009), 1411-1420.)

2) As $c$ tends to infinity, the Hausdorff dimension of the Julia set tends to zero. This is because the Julia set can be written as the limit set of a conformal iterated function system with two maps, corresponding to the inverse branches of the maps, and these are strongly contracting if $c$ is large.

3) As Alex mentions, Hausdorff dimension does not vary continuously for parameters on the boundary of the Mandelbrot set. In fact, the following is true:

Theorem. Suppose that $c\in \partial M$. Then there is a sequence $(c_n)$ of parameters outside the Mandelbrot set such that $\dim(J(q_{c_n}))\to 2$.

This follows from Shishikura's famous proof that the boundary of the Mandelbrot set has Hausdorff dimension equal to $2$ ("The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets", Ann. of Math. 147 (1998), no. 2, 225–267). Indeed, he shows that there is a dense set of parameters on the boundary where the hyperbolic dimension equals two. Any nearby parameter will have a Julia set of Hausdorff dimension close to $2$.

On the other hand, there are many parameters on the boundary of the Mandelbrot set where the Hausdorff dimension is strictly less than $2$. So it is not hard to see that the dimension does not depend continuously in the way that you desire.

If we ask about radial limits (i.e., consider the conformal map that takes the complement of the closed unit disk to the complement of the Mandelbrot set, and approach the boundary of the Mandelbrot set along the image of a straight ray), things become more subtle, and I am not sure what exactly is known. However, from what I can remember, it is known that, even for the simple case where $c(t) = 1/4+t$, $t>0$, the Hausdorff dimension of $J(q_{c(t)})$ does not tend to that of $J(q_{1/4})$ as $t\to 0$. (This is the parabolic implosion that Alex mentions.)

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At some points of the boundary of M, the Hausdorff dimension is not continuous, these are points of "parabolic implosion", see MR2521938.

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