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arXiv · 1710.02371

Dynamics of the scenery flow and conical density theorems

Antti Käenmäki
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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.

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Antti Käenmäki. 2017-10-06. Dynamics of the scenery flow and conical density theorems. https://doi.org/10.4064/bc115-4

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