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 (FD) in the sense of Hochman is generated by a USM, 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 FDs 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 FDs as tangent distributions.