Dynamics of the scenery flow and geometry of measures

Abstract

We employ the ergodic theoretic machinery of scenery flows to address classical geometric measure theoretic problems on Euclidean spaces. Our main results include a sharp version of the conical density theorem, which we show to be closely linked to rectifiability. Moreover, we show that the dimension theory of measure-theoretical porosity can be reduced back to its set-theoretic version, that Hausdorff and packing dimensions yield the same maximal dimension for porous and even mean porous measures, and that extremal measures exist and can be chosen to satisfy a generalized notion of self-similarity. These are sharp general formulations of phenomena that had been earlier found to hold in a number of special cases.

v3: 30 pages, 2 figures, fixed typos and minor errors, to appear in Proc. London Math. Soc

References in corpus (8)

• Packing dimension and Ahlfors regularity of porous sets in metric spaces
• Conical upper density theorems and porosity of measures
• Nonsymmetric conical upper density and $k$-porosity
• Directed porosity on conformal iterated function systems and weak convergence of singular integrals
• Upper Conical density results for general measures on ${\mathbb{R}}^n$
• On upper conical density results
• Structure of distributions generated by the scenery flow
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Cited by in corpus (9)

• Weak separation condition, Assouad dimension, and Furstenberg homogeneity
• On the Hausdorff dimension of microsets
• Self-affine sets with fibered tangents
• Locally rich compact sets
• Structure of distributions generated by the scenery flow
• On distance sets, box-counting and Ahlfors-regular sets
• Micromeasure distributions and applications for conformally generated fractals
• Dynamics of the scenery flow and conical density theorems
• Scenery flow, conical densities, and rectifiability