Local dimensions of measures of finite type

Asian Journal of Mathematics

Matthews

2019

Self Cite

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 R that are of finite type.In this paper, our focus is on finite type measures defined on the torus, the quotient space 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 selfsimilar 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 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 R.

Section: 3mentioning

confidence: 99%

Section: Symbolic Representations and The Essential Class(es)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Cantor-like measures are of finite type on R (c.f. [9,11]). In [11] this fact was used to study their local dimensions, extending earlier work of [1,12,15].…”

mentioning

confidence: 99%

See 3 more Smart Citations

Local dimensions of measures of finite type on the torus

Asian Journal of Mathematics

Matthews

2019

Self Cite

Local dimensions of self-similar measures satisfying the finite neighbour condition*

Rutar

2022

Nonlinearity

We study sets of local dimensions for self-similar measures in R satisfying the finite neighbour condition, which is formally stronger than the weak separation condition (WSC) 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 contextualises and generalises a vast amount of prior work on sets of local dimensions for self-similar measures satisfying the WSC.

Local Dimensions of Measures of Finite Type II: Measures Without Full Support and With Non-regular Probabilities

2018

Can. j. math.

Self Cite

Consider a sequence of linear contractions S j (x) = ̺x + d j and probabilities p j > 0 with p j = 1. We are interested in the self-similar measure µ = p j µ • S −1 j , of finite type. In this paper we study the multifractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval.Under some mild technical assumptions, we prove that there exists a subset of suppµ of full µ and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff measure of their support and we show that the dimension of the support can be computed using only information about the essential class.To conclude, we present a detailed study of three examples. First, we show that the set of local dimensions of the biased Bernoulli convolution with contraction ratio the inverse of a simple Pisot number always admits an isolated point. We give a precise description of the essential class of a generalized Cantor set of finite type, and show that the kth convolution of the associated Cantor measure has local dimension at x ∈ (0, 1) tending to 1 as k tends to infinity. Lastly, we show that within a maximal loop class that is not truly essential, the set of upper local dimensions need not be an interval. This is in contrast to the case for finite type measures with regular probabilities and full interval support.