Local Geometry of Self-similar Sets: Typical Balls, Tangent Measures and Asymptotic Spectra

TL;DR

This paper investigates the local geometric structure of self-similar sets, revealing the complexity and uniformity properties of typical spherical neighborhoods and tangent measures, with explicit spectrum calculations.

Contribution

It introduces a detailed analysis of typical balls in self-similar sets, characterizes their tangent measures, and computes the asymptotic density spectrum for specific fractals.

Findings

Uncountably many classes of spherical neighborhoods are not equivalent under similitudes.

Any typical ball is a tangent measure at almost every point for self-similar measures.

Computed the spectrum of asymptotic densities for the Sierpinski gasket.

Abstract

We analyse the local geometric structure of self-similar sets with open set condition through the study of the properties of a distinguished family of spherical neighbourhoods, the typical balls. We quantify the complexity of the local geometry of self-similar sets, showing that there are uncountably many classes of spherical neighbourhoods that are not equivalent under similitudes. We show that, at a tangent level, the uniformity of the Euclidean space is recuperated in the sense that any typical ball is a tangent measure of the measure $\nu $ at $\nu $-a.e. point, where $\nu $ is any self-similar measure. We characterise the spectrum of asymptotic densities of metric measures in terms of the packing and centred Hausdorff measures. As an example, we compute the spectrum of asymptotic densities of the Sierpinski gasket.