How to find the Hausdorff Dimension of the Julia set of $z^2+2$, which is $J=J(z^2+2)$?
I am recently doing Fractal Geometry from a book by Kenneth Falconer. The book gives some estimation about the Hausdorff dimension of $z^2+c$ where $c$ is very large or very small. But $2$ is neither big nor small. So I cannot estimate its Hausdorff dimension $s$ and the corresponding Hausdorff measure $\mathcal{H}^s(J)$.
By the way, I dislike one aspect of the book. It includes a little too much description in English. I think if the author uses mathematical notations instead of heavy English prose, I would have less difficulties translating English to Mathematics. Is my opinion appropriate?
Or is it too difficult for human to answer this question?
Or just give me a numerical approximation
I am fairly certain that there is no explicit formula for the Hausdorff dimension of the Julia set of $z^2+2$. In fact, except for some highly special parameters such as $z^2$ or $z^2-2$ there is no explicit formula.
The best that one can hope for is either an algorithm giving you a numerical approximation, or asymptotic bounds near those special parameters or near infinity.
Note that just because there is no explicit formula, it doesn't mean that one cannot ask interesting questions about the Hausdorff dimension $\delta(c)$ of Julia sets of $z^2+c$. Here's a couple that are (to my knowledge) open, though some partial answers are known:
at which $c$ is $\delta(c)$ continuous?
at which $c$ does $\delta(c)$ attain a minimum?
is it true that $\delta$ is decreasing between $c=-1.41...$ (Feigenbaum point) and $c=0$?