Local dimensions of measures of finite type

TL;DR

This paper investigates the multifractal structure of certain self-similar measures of finite type, introducing a combinatorial framework to analyze local dimensions and revealing their distribution and density properties.

Contribution

It introduces the concept of loop classes to analyze local dimensions of finite type measures, providing formulas and density results for periodic points and examples with unique local dimension structures.

Findings

Set of attainable local dimensions forms a closed interval within a positive loop class.

Local dimensions at periodic points are dense and have a simple formula.

Some measures have a single interval of local dimensions with isolated points, or exactly two distinct local dimensions.

Abstract

We study the multifractal analysis of a class of equicontractive, self-similar measures of finite type, whose support is an interval. Finite type is a property weaker than the open set condition, but stronger than the weak open set condition. Examples include Bernoulli convolutions with contraction factor the inverse of a Pisot number and self-similar measures associated with $m$-fold sums of Cantor sets with ratio of dissection $1/R$ for integer $R\leq m$. We introduce a combinatorial notion called a loop class and prove that the set of attainable local dimensions of the measure at points in a positive loop class is a closed interval. We prove that the local dimensions at the periodic points in the loop class are dense and give a simple formula for those local dimensions. These self-similar measures have a distinguished positive loop class called the essential class. The set of points in the essential class has full Lebesgue measure in the support of the measure and is often all but the two endpoints of the support. Thus many, but not all, measures of finite type have at most one isolated point in their set of local dimensions. We give examples of Bernoulli convolutions whose sets of attainable local dimensions consist of an interval together with an isolated point. As well, we give an example of a measure of finite type that has exactly two distinct local dimensions.