Dynamics of the scenery flow and geometry of measures
Käenmäki, A., Sahlsten, T., & Shmerkin, P. (2015). Dynamics of the scenery flow and geometry of measures. Proceedings of the London mathematical society, 110 (5), 1248-1280. doi:10.1112/plms/pdv003
Published inProceedings of the London mathematical society
© London Mathematical Society. This is a final draft version of an article whose final and definitive form has been published by London Mathematical Society.
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.