Title: Dynamics of the scenery flow and conical density theorems,
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.