Local Dimensions of Self-similar Measures Satisfying the Finite Neighbour Condition

TL;DR

This paper investigates the structure of local dimensions for self-similar measures satisfying the finite neighbour condition, showing they form a finite union of compact intervals and providing methods for explicit computation.

Contribution

It establishes that the set of local dimensions is a finite union of intervals under the finite neighbour condition, generalizing prior results and enabling explicit calculations.

Findings

Set of local dimensions is a finite union of compact intervals.

Number of intervals bounded by the structure of a directed graph.

Provides methods for explicit computation of local dimensions.

Abstract

We study sets of local dimensions for self-similar measures in $\mathbb{R}$ satisfying the finite neighbour condition, which is formally stronger than the weak separation condition but satisfied in all known examples. Under a mild technical assumption, we establish that the set of attainable local dimensions is a finite union of (possibly singleton) compact intervals. The number of intervals is bounded above by the number of non-trivial maximal strongly connected components of a finite directed graph construction depending only on the governing iterated function system. We also explain how our results allow computations of the sets of local dimensions in many explicit cases. This contextualizes and generalizes a vast amount of prior work on sets of local dimensions for self-similar measures satisfying the weak separation condition.