Tangent measures of self-similar sets satisfying the strong separation condition

TL;DR

This paper provides an explicit formula for tangent measures of self-similar sets satisfying the strong separation condition, revealing their structure through zooming-in dynamics and extending results to sets under the open set condition.

Contribution

It derives a uniform formula for tangent measures of self-similar sets with strong separation and characterizes tangent measures for sets under the open set condition.

Findings

Explicit tangent measure formula for strong separation sets

Support of tangent measures corresponds to limit models

Extension of tangent measure characterization to open set condition sets

Abstract

This paper investigates tangent measures in the sense of Preiss for self-similar sets on ${{\mathbb{R}}^d}$ that satisfy the strong separation condition. Through the dynamics of ``zooming in'' on any typical point, we derive an explicit and uniform formula for the tangent measures associated with this category of self-similar sets on ${{\mathbb{R}}^d}$. Furthermore, for any self-similar set $C\subset{{\mathbb{R}}^d}$ under the open set condition instead of the strong separation condition, we find that the support of any tangent measure at each point $x\in C$ is one of the limit models at that point. Conversely, any limit model at each point $x\in C$ is the support of one of the tangent measures at that point.