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.
Main Authors
Format
Articles
Research article
Published
2015
Series
Subjects
Publication in research information system
Publisher
Oxford University Press; London Mathematical Society
The permanent address of the publication
https://urn.fi/URN:NBN:fi:jyu-201510213435Use this for linking
Review status
Peer reviewed
ISSN
0024-6115
DOI
https://doi.org/10.1112/plms/pdv003
Language
English
Published in
Proceedings of the London mathematical society
Citation
- 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. https://doi.org/10.1112/plms/pdv003
License
Copyright© London Mathematical Society. This is a final draft version of an article whose final and definitive form has been published by London Mathematical Society.