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

Yongtao Wang

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.

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 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 . Furthermore, for any self-similar set under the open set condition instead of the strong separation condition, we find that the support of any tangent measure at each point is one of the limit models at that point. Conversely, any limit model at each point is the support of one of the tangent measures at that point.
Paper Structure (9 sections, 23 theorems, 187 equations)

This paper contains 9 sections, 23 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of limit models in ${{\mathbb{R}}^d}$
  4. General definition and facts in ${{\mathbb{R}^d}}$
  5. Limit models of self-similar sets in ${{\mathbb{R}}^d}$
  6. Proof of Theorem 1.1
  7. Proof of Theorem \ref{['th:1.2']}
  8. $\omega({C^x}) \supset \{\operatorname{spt}\nu|\nu \in {\mathop{\rm Tan}\nolimits} (\mu ,x)\}$
  9. $\omega({C^x}) = \{\operatorname{spt} \nu|\nu \in {\mathop{\rm Tan}\nolimits} (\mu ,x)\}$

Key Result

theorem 1

Let $C\subset{{\mathbb{R}^d}}$ be a self-similar set satisfying the strong separation condition (see Definition de:2.2 and Definition de:2.4). Let $\mu= {{\cal H}^s}\lfloor_C$ with $s={\dim_H}(C)$. Then for any $x \in C$, we have where $\omega({C^x})$ denotes the class of "tangent set" of $C$ at $x$ (see Definition de:3.1).

Theorems & Definitions (43)

  • theorem 1
  • remark 1
  • theorem 2
  • remark 2
  • definition 1: ref10; 4.13.Self-similar sets
  • definition 2: see ref7ref1
  • definition 3
  • definition 4
  • definition 5
  • remark 3
  • ...and 33 more