Abstract
This is a survey on multifractal analysis with an emphasis on the multifractal geometry of geometric constructions involving (multifractal) measures.
We first describe the basic notions in multifractal analysis: the Multifractal Formalism, the coarse multifractal spectra, the box (Legendre) multifractal spectra, the fine multifractal spectra, and discuss the relationship between these multifractal spectra (Sections 1-3).
We then (Section 4) consider general geometric constructions in multifractal geometry. We show that the fine multifractal formalism introduced by Olsen [Ol11], Pesin [Pes1, Pes2] and Peyriere [Pey] leads to a multifractal geometry for product measures, for slices of measures (i.e. intersections of measures with lower dimensional subspaces), and for general intersections of measures, which is completely analogous to the fractal geometry for product sets, for slices of sets (i.e. intersections of sets with lower dimensional subspaces), and for general intersections of sets, respectively.
| Original language | English |
|---|---|
| Publication status | Published - 2000 |
Keywords
- fractals
- multifractals
- multifractal spectrum
- coarse multifractal spectrum
- fine multifractal spectrum
- Hausdorff dimension
- packing dimension
- slices of measures
- intersections of measures
- SELF-SIMILAR MEASURES
- ITERATED RANDOM MULTIPLICATIONS
- STABLE OCCUPATION MEASURE
- AXIOM-A DIFFEOMORPHISMS
- DIMENSION SPECTRUM
- STRANGE ATTRACTORS
- FOURIER-TRANSFORMS
- FRACTAL MEASURES
- THERMODYNAMIC FORMALISM
- INVARIANT DISTRIBUTIONS