A continuous translation based method to reveal the fine structure of fractal

Abstract EN

We introduce a new method based on Hausdorff Measure Spectrum Function (HMSF) which provides a more precise way for tracing the geometrical organization of a fractal set.

We use Cantor sets as a tractable fractal model.

Based on the triadic expansion of the shifting number, we first determine the fractal structure of intersection of triadic Cantor sets with their translates.

We found that the Hausdorff measure of these sets forms a discrete spectrum whose non-zero values come only from those shifting numbers with a finite triadic expansion.

This spectrum does carry a huge amount of information about the set to likely be explored in a chosen way.

To extract this information, a one by one step method is developed and applied to typical fractal sets.

This results in a complete identification of fractals.