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Dynamics of the scenery flow and conical density theorems

Volume 115 / 2018

Antti Käenmäki Banach Center Publications 115 (2018), 99-143 MSC: Primary 28A80; Secondary 37A10, 28A75, 28A33. DOI: 10.4064/bc115-4

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.

Authors

  • Antti KäenmäkiDepartment of Mathematics and Statistics
    P.O. Box 68 (Gustaf Hälströmin katu 2b)
    FI-00014 University of Helsinki
    Finland
    e-mail

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