Local dimensions of measures of finite type II - Measures without full support and with non-regular probabilities

Hare, Kathryn E.; Ng, Michael Ka Shing

doi:10.48550/arxiv.1603.02244

Cited by 1 publication

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“…is the other net interval of level n with x as endpoint. Since all quotient net intervals of level n have lengths comparable to ρ n one can see in the same manner as in [11,Thm. 2.6] that…”

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

confidence: 76%

“…Remark 2.11. In [11,Proposition 3.6] it was shown that, for self-similar measures of finite type on R, the set of points in the essential class has full µ-measure, and full H s -measure. Here H s is the Hausdorff measure associated to the support of µ.…”

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

confidence: 99%

“…The proof is the same as in [11,Proposition 2.7] as that argument only required the formula we have developed for µ π (∆) and the fact that the primitive transition matrices have a non-zero entry in each column.…”

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

confidence: 99%

“…In [3,4,5], Feng conducted a detailed study of equicontractive, self-similar measures of finite type on R, focussing primarily on Bernoulli convolutions with contraction factors the inverse of a simple Pisot number. In [9,11], the authors, together with Ng, showed that if µ is a measure of finite type on R that arises from regular probabilities and has full interval support, then the set of local dimensions of µ at the points in any positive loop class is a closed interval and the local dimensions at periodic points in the loop class are dense within this interval. See Section 2.1 for a definition of regular probabilities.…”

Section: Introductionmentioning

confidence: 99%

“…Measures which satisfy the open set condition are of finite type on R and measures of finite type on R satisfy the weak open set condition [14]. The structure of finite type measures and aspects of their multi-fractal analysis is explained in detail in [3,4,5,9,11].…”

Section: Introductionmentioning

confidence: 99%

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Local dimensions of measures of finite type on the torus

Hare

Matthews

2019

Asian Journal of Mathematics

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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.

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

confidence: 76%

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

confidence: 99%

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

confidence: 99%