Towards a Noncommutative Fractal Geometry. Laplacians and Volume Measures onFractals

摘要

We review some recent work of the author (and his collaborators) regardingharmonic analysis and partial differential equations on fractals; more specifically, the vibrations of 'fractal drums' and the spectral distribution of Laplacians on (suitable) self-similar fractals. We also discuss how this work was combined by the author with techniques from Connes' noncommutative geometry to construct 'volume measures' on such fractals, including an analogue in this context of the Riemannian volume measure. Further, we announce new results regarding the nature of these volume measures and, in special cases. their relationship with a suitable notion of Hausdorff measure. In addition, we consider the notion of 'complex dimension' of (self-similar) fractals and its relationship with oscillatory phenomena in our setting.