Generic Hölder level sets on fractals

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