Dynamics of the scenery flow and geometry of measures
Abstract
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.
Keywords
Cite
@article{arxiv.1401.0231,
title = {Dynamics of the scenery flow and geometry of measures},
author = {Antti Käenmäki and Tuomas Sahlsten and Pablo Shmerkin},
journal= {arXiv preprint arXiv:1401.0231},
year = {2017}
}
Comments
v3: 30 pages, 2 figures, fixed typos and minor errors, to appear in Proc. London Math. Soc