Elements of the proof of Theorem 1.5 and Proposition 1.6

Abstract

The spectrum of singularities of X _t coincides with that of Z. Consequently, to prove Theorem 1.5, we have to determine Hausdorff dimensions of the sets

$$\displaystyle\begin{array}{rcl} \mathcal{E}_{Z,\eta }&:=& \{x \in (0,1): H_{Z}(x) =\eta \}, {}\\ \tilde{\mathcal{E}}_{Z,\eta }&:=& \{x \in (0,1): H_{Z}(x) \leq \eta \} {}\\ \end{array}$$

and this is done in the next two sections.