The scenery flow of self-similar measures with weak separation condition

TL;DR

This paper proves that self-similar measures on Euclidean spaces satisfying the weak separation condition are uniformly scaling, using ergodic theory and geometric analysis to understand their structure.

Contribution

It establishes the uniform scaling property for self-similar measures under the weak separation condition, combining ergodic theory with geometric analysis.

Findings

Self-similar measures with weak separation are uniformly scaling.

The approach integrates ergodic theory and geometric analysis.

Provides a new understanding of measure structure under weak separation.

Abstract

We show that self-similar measures on $\mathbb{R}^d$ satisfying the weak separation condition are uniformly scaling. Our approach combines elementary ergodic theory with geometric analysis of the structure given by the weak separation condition.