Dynamics of the scenery flow and conical density theorems

TL;DR

This paper explores how the scenery flow, a dynamical system derived from measure blow-ups, can be used with ergodic theory to analyze conical density theorems in geometric measure theory.

Contribution

It develops ergodic-theoretical methods around the scenery flow to advance understanding of conical density theorems.

Findings

Application of ergodic theory to conical densities

Development of scenery flow techniques

Enhanced understanding of measure distribution in small balls

Abstract

Conical density theorems are used in the geometric measure theory to derive geometric information from given metric information. The idea is to examine how a measure is distributed in small balls. Finding conditions that guarantee the measure to be effectively spread out in different directions is a classical question going back to Besicovitch (1938) and Marstrand (1954). Classically, conical density theorems deal with the distribution of the Hausdorff measure. The process of taking blow-ups of a measure around a point induces a natural dynamical system called the scenery flow. Relying on this dynamics makes it possible to apply ergodic-theoretical methods to understand the statistical behavior of tangent measures. This approach was initiated by Furstenberg (1970, 2008) and greatly developed by Hochman (2010). The scenery flow is a well-suited tool to address problems concerning conical densities. In this survey, we demonstrate how to develop the ergodic-theoretical machinery around the scenery flow and use it to study conical density theorems.