We employ the ergodic theoretic machinery of scenery flows to address classical geometric measure theoretic problems on Euclidean spaces. Our main results include a sharp version of the conical density theorem, which we show to be closely linked to rectifiability. Moreover, we show that the dimension theory of measure-theoretical porosity can be reduced back to its set-theoretic version, that Hausdorff and packing dimensions yield the same maximal dimension for porous and even mean porous measures, and that extremal measures exist and can be chosen to satisfy a generalized notion of self-similarity. These are sharp general formulations of phenomena that had been earlier found to hold in a number of special cases.
- Pub Date:
- December 2013
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Dynamical Systems;
- Primary 28A80;
- Secondary 37A10;
- v3: 30 pages, 2 figures, fixed typos and minor errors, to appear in Proc. London Math. Soc