Section: New Results

Local strings and the CH set

Participant : Jacques Lévy Véhel.

In collaboration with Prof. Franklin Mendivil (Acadia University, Canada).

We have extended the definition of fractal strings originally proposed in [59] and modified in [37] to deal with the local behaviour of fractal sets. This allows to analyze the pointwise oscillatory properties of locally self-similar sets ([38] ).

We have also analyzed in details the structure of a set build by "stacking" Cantor sets with continuously varying dimensions (see figure 4 ). The resulting set, called "Christiane's hair" set or CH set, displays a number of interesting properties. Each "strand of hair" is a $C^{\infty }$ curve. Its Hausdorf dimension is 2. Furthermore, it is Minkowski measurable in dimension 2 with vanishing Minkowski content.

Figure 4. The CH set.