Structure of distributions generated by the scenery flow
Abstract
We expand the ergodic theory developed by Furstenberg and Hochman on dynamical
systems that are obtained from magnifications of measures. We prove that any fractal distribution
in the sense of Hochman is generated by a uniformly scaling measure, which provides a converse to
a regularity theorem on the structure of distributions generated by the scenery flow. We further
show that the collection of fractal distributions is closed under the weak topology and, moreover, is
a Poulsen simplex, that is, extremal points are dense. We apply these to show that a Baire generic
measure is as far as possible from being uniformly scaling: at almost all points, it has all fractal
distributions as tangent distributions.
Main Authors
Format
Articles
Research article
Published
2015
Series
Subjects
Publication in research information system
Publisher
Oxford University Press
The permanent address of the publication
https://urn.fi/URN:NBN:fi:jyu-201510213437Use this for linking
Review status
Peer reviewed
ISSN
0024-6107
DOI
https://doi.org/10.1112/jlms/jdu076
Language
English
Published in
Journal of the London Mathematical Society
Citation
- Käenmäki, A., Sahlsten, T., & Shmerkin, P. (2015). Structure of distributions generated by the scenery flow. Journal of the London Mathematical Society, 91(2), 464-494. https://doi.org/10.1112/jlms/jdu076
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.