Dynamics of the scenery flow and conical density theorems
| Autor: | Antti Käenmäki |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
History and Overview (math.HO)
Mathematics - History and Overview Mathematical analysis Conical surface Dynamical Systems (math.DS) Measure (mathematics) 28A80 37A10 28A75 28A33 Geometric measure theory Flow (mathematics) Tangent measure Metric (mathematics) FOS: Mathematics General Earth and Planetary Sciences Hausdorff measure Mathematics - Dynamical Systems Dynamical system (definition) General Environmental Science Mathematics |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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