Dynamics on fractals and fractal distributions

TL;DR

This paper explores the dynamics of fractal measures through zooming in on points, analyzing their limiting distributions and geometric properties, and relating them to existing models like Z"ahle distributions and Furstenberg's CP-processes.

Contribution

It develops foundational properties of these limiting distributions and investigates their geometric behavior under projection and conditioning.

Findings

Limiting distributions often equidistribute for invariant measures

Relations established between fractal measures and Z"ahle distributions

Geometric properties under projection and conditioning analyzed

Abstract

We study fractal measures on Euclidean space through the dynamics of "zooming in" on typical points. The resulting family of measures (the "scenery"), can be interpreted as an orbit in an appropriate dynamical system which often equidistributes for some invariant distribution. The first part of the paper develops basic properties of these limiting distributions and the relations between them and other models of dynamics on fractals, specifically to Z\"ahle distributions and Furstenberg's CP-processes. In the second part of the paper we study the geometric properties of measures arising in these contexts, specifically their behavior under projection and conditioning on subspaces.