  Hausdorff dimensions of level sets of generic continuous functions defined on
fractals were considered in two papers by R. Balka, Z. Buczolich and M. Elekes.
In those papers the topological Hausdorff dimension of fractals was defined. In
this paper we start to study level sets of generic $1$-H\"older-$\alpha$
functions defined on fractals. This is related to some sort of "thickness",
"conductivity" properties of fractals. The main concept of our paper is
$D_{*}(\alpha, F)$ which is the essential supremum of the Hausdorff dimensions
of the level sets of a generic $1$-H\"older-$\alpha$ function defined on the
fractal $F$. We prove some basic properties of $D_{*}(\alpha, F)$, we calculate
its value for an example of a "thick fractal sponge", we show that for
connected self similar sets $D_{*}(\alpha, F)$ it equals the Hausdorff
dimension of almost every level in the range of a generic $1$-H\"older-$\alpha$
function.
