Local dimensions of measures of finite type on the torus

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 $\mathbb{R/Z}$. 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 $\mathbb{R}$.

Keywords

multi-fractal analysis, local dimension, IFS, finite type, quotient space

2010 Mathematics Subject Classification

11R06, 28A78, 28A80

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The research of K. E. Hare and K. R. Matthews was supported by NSERC Grant 44597-2011. The research of K. G. Hare was supported by NSERC Grant RGPIN-2014-03154.

Received 18 July 2016

Accepted 27 July 2017

Published 3 May 2019