The topology and combinatorial structure of the Mandelbrot set ${ℳ}^d$ (of degree d ≥ 2) can be studied using symbolic dynamics. Each parameter is mapped to a kneading sequence, or equivalently, an internal address; but not every such sequence is realized by a parameter in ${ℳ}^d$. Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, ${\Lambda }^d$. In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence is realized....

In the moduli space ${ℳ}_d$ of degree $d$ rational maps, the bifurcation locus is the support of a closed $(1,1)$ positive current $T_{\mathrm{bif}}$ which is called the bifurcation current. This current gives rise to a measure ${\mu }_{\mathrm{bif}}:={\left(T_{\mathrm{bif}}\right)}^{2d-2}$ whose support is the seat of strong bifurcations. Our main result says that $\mathrm{supp}\left({\mu }_{\mathrm{bif}}\right)$ has maximal Hausdorff dimension $2(2d-2)$. As a consequence, the set of degree $d$ rational maps having $(2d-2)$ distinct neutral cycles is dense in a set of full Hausdorff dimension.