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

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.