Structure of distributions generated by the scenery flow

Antti Käenmäki, Tuomas Sahlsten, Pablo Shmerkin

Published 2013-12-09, updated 2015-03-05Version 2

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.