Local dimensions of measures of finite type on the torus

TL;DR

This paper investigates the local dimensions of finite type measures on the torus, establishing conditions for interval sets, constructing examples with isolated points, and contrasting these properties with measures on the real line.

Contribution

It provides criteria for when the local dimensions form an interval, constructs examples with isolated points, and analyzes convolutions of Cantor-like measures on the torus.

Findings

Set of local dimensions of certain measures on the torus can be an interval.

Existence of measures on the torus with isolated points in their local dimensions.

Convolutions of Cantor-like measures on the torus typically lack isolated points.

Abstract

The structure of the set of local dimensions of a self-similar measure has been studied by numerous mathematicians, initially for measures that satisfy the open set condition and, more recently, for measures on $\mathbb{R}$ that are of finite type. In this paper, our focus is on finite type measures defined on the torus, the quotient space $\mathbb{R}\backslash \mathbb{Z}$. We give criteria which ensures that the set of local dimensions of the measure taken over points in special classes generates an interval. We construct a non-trivial example of a measure on the torus that admits an isolated point in its set of local dimensions. We prove that the set of local dimensions for a finite type measure that is the quotient of a self-similar measure satisfying the strict separation condition is an interval. We show that sufficiently many convolutions of Cantor-like measures on the torus never admit an isolated point in their set of local dimensions, in stark contrast to such measures on $\mathbb{R}$. Further, we give a family of Cantor-like measures on the torus where the set of local dimensions is a strict subset of the set of local dimensions, excluding the isolated point, of the corresponding measures on $\mathbb{R}$.