scholarly journals Lower Assouad Dimension of Measures and Regularity

2019 ◽  
pp. 1-37 ◽  
Author(s):  
KATHRYN E. HARE ◽  
SASCHA TROSCHEIT
Keyword(s):  
Lower Bounds ◽  
Open Set Condition ◽  
Local Behaviour ◽  
Pisot Numbers ◽  
Open Set ◽  
Regularity Properties ◽  
Assouad Dimension ◽  
Uniformly Perfect ◽  
Bernoulli Convolutions ◽  
Self Similar

Abstract In analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular, the quasi-lower Assouad dimension is dominated by the infimum of the measure’s lower local dimensions. Although strict inequality is possible in general, equality holds for the class of self-similar measures of finite type. This class includes all self-similar, equicontractive measures satisfying the open set condition, as well as certain “overlapping” self-similar measures, such as Bernoulli convolutions with contraction factors that are inverses of Pisot numbers. We give lower bounds for the lower Assouad dimension for measures arising from a Moran construction, prove that self-affine measures are uniformly perfect and have positive lower Assouad dimension, prove that the Assouad spectrum of a measure converges to its quasi-Assouad dimension and show that coincidence of the upper and lower Assouad dimension of a measure does not imply that the measure is s-regular.

scholarly journals The quasi-Assouad dimension of stochastically self-similar sets

2019 ◽  
Vol 150 (1) ◽  
pp. 261-275 ◽  
Author(s):  
Sascha Troscheit
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function Systems ◽  
Random Sets ◽  
Open Set Condition ◽  
Fractal Percolation ◽  
Open Set ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar

AbstractThe class of stochastically self-similar sets contains many famous examples of random sets, for example, Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive sets is almost surely equal to the almost sure Hausdorff dimension of the set. We further comment on random homogeneous and V -variable sets and the removal of overlap conditions.

scholarly journals Inhomogeneous self-similar sets with overlaps

2017 ◽  
Vol 39 (1) ◽  
pp. 1-18 ◽  
Author(s):  
SIMON BAKER ◽  
JONATHAN M. FRASER ◽  
ANDRÁS MÁTHÉ
Keyword(s):  
Final Section ◽  
Iterated Function System ◽  
Box Dimension ◽  
Open Set Condition ◽  
Open Set ◽  
Bernoulli Convolutions ◽  
Iterated Function ◽  
Different Types ◽  
Self Similar ◽  
Weak Separation

It is known that if the underlying iterated function system satisfies the open set condition, then the upper box dimension of an inhomogeneous self-similar set is the maximum of the upper box dimensions of the homogeneous counterpart and the condensation set. First, we prove that this ‘expected formula’ does not hold in general if there are overlaps in the construction. We demonstrate this via two different types of counterexample: the first is a family of overlapping inhomogeneous self-similar sets based upon Bernoulli convolutions; and the second applies in higher dimensions and makes use of a spectral gap property that holds for certain subgroups of $\text{SO}(d)$ for $d\geq 3$. We also obtain new upper bounds, derived using sumsets, for the upper box dimension of an inhomogeneous self-similar set which hold in general. Moreover, our counterexamples demonstrate that these bounds are optimal. In the final section we show that if the weak separation property is satisfied, that is, the overlaps are controllable, then the ‘expected formula’ does hold.

ON THE ASSOUAD DIMENSION OF GRAPH DIRECTED MORAN FRACTALS

Fractals ◽  
2011 ◽  
Vol 19 (02) ◽  
pp. 221-226 ◽  
Author(s):  
L. OLSEN
Keyword(s):  
Direct Proof ◽  
Open Set Condition ◽  
Open Set ◽  
Assouad Dimension ◽  
Box Dimensions

We give a simple and direct proof of the fact that the Assouad dimension of a graph directed Moran fractal satisfying the Open Set Condition coincides with its Hausdorff and box dimensions.

Open set condition for graph directed self-similar structure

2013 ◽  
Vol 276 (1-2) ◽  
pp. 243-260 ◽  
Author(s):  
Tian-jia Ni ◽  
Zhi-ying Wen
Keyword(s):  
Open Set Condition ◽  
Open Set ◽  
Self Similar

scholarly journals SPACE-FILLING CURVES OF SELF-SIMILAR SETS (III): SKELETONS

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050028
Author(s):  
HUI RAO ◽  
SHU-QIN ZHANG
Keyword(s):  
Finite Type ◽  
Type Condition ◽  
Space Filling ◽  
Open Set Condition ◽  
Substitution Rule ◽  
Space Filling Curves ◽  
Neighbor Graph ◽  
Open Set ◽  
Self Similar

Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. In a previous paper by Dai and the authors [Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule, Nonlinearity 32(5) (2019) 1772–1809] it was shown that for all the connected self-similar sets with a skeleton satisfying the open set condition, space-filling curves can be constructed. In this paper, we give a criterion of existence of skeletons by using the so-called neighbor graph of a self-similar set. In particular, we show that a connected self-similar set satisfying the finite-type condition always possesses skeletons: an algorithm is obtained here.

scholarly journals Lattice self-similar sets on the real line are not Minkowski measurable

2018 ◽  
Vol 40 (1) ◽  
pp. 221-232
Author(s):  
SABRINA KOMBRINK ◽  
STEFFEN WINTER
Keyword(s):  
Real Line ◽  
Iterated Function System ◽  
Function System ◽  
Open Set Condition ◽  
The Real ◽  
Open Set ◽  
Puzzle Piece ◽  
Iterated Function ◽  
Self Similar ◽  
Special Classes

We show that any non-trivial self-similar subset of the real line that is invariant under a lattice iterated function system (IFS) satisfying the open set condition (OSC) is not Minkowski measurable. So far, this has only been known for special classes of such sets. Thus, we provide the last puzzle-piece in proving that under the OSC a non-trivial self-similar subset of the real line is Minkowski measurable if and only if it is invariant under a non-lattice IFS, a 25-year-old conjecture.

Open set condition and post-critically finite self-similar sets

Nonlinearity ◽  
2008 ◽  
Vol 21 (6) ◽  
pp. 1227-1232 ◽  
Author(s):  
Qi-Rong Deng ◽  
Ka-Sing Lau
Keyword(s):  
Open Set Condition ◽  
Open Set ◽  
Self Similar

Empirical multifractal moment measures and moment scaling functions of self-similar multifractals

2002 ◽  
Vol 133 (3) ◽  
pp. 459-485 ◽  
Author(s):  
L. OLSEN
Keyword(s):  
Lipschitz Constant ◽  
The Self ◽  
Probability Vector ◽  
Periodic Functions ◽  
Open Set Condition ◽  
Similar Measure ◽  
Open Set ◽  
Moment Measures ◽  
Self Similar ◽  
Exact Rate Of Convergence

Let Si: ℝd → ℝd for i = 1, …, n be contracting similarities, and let (p1, …, pn) be a probability vector. Let K and μ be the self-similar set and the self-similar measure associated with (Si,pi)i. For q ∈ ℝ and r > 0, define the qth covering moment and the qth packing moment of μ by[formula here]where the infimum is taken over all r-spanning subsets E of K, and the supremum is taken over all r-separated subsets F of K. If the Open Set Condition (OSC) is satisfied then it is well known that[formula here]where β(q) is defined by [sum ]ipqirβi(q) = 1 (here ri denotes the Lipschitz constant of Si). Assuming the OSC, we determine the exact rate of convergence in (*): there exist multiplicatively periodic functions πq, Πq: (0,∞) → ℝ such that[formula here]where ε(r) → 0 as r[searr ]0. As an application of (**) we show that the empirical multi-fractal moment measures converges weakly:[formula here]where, for each positive r, Er is a (suitable) minimal r-spanning subset of K and Fr is a (suitable) maximal r-separated subset of K, and [Hscr ]q,β(q)μ and [Pscr ]q,β(q)μ are the multifractal Hausdorff measure and the multifractal packing measure, respectively.

scholarly journals The mutual singularity of the relative multifractal measures

2021 ◽  
Vol 8 (1) ◽  
pp. 18-26
Author(s):  
Zied Douzi ◽  
Bilel Selmi
Keyword(s):  
General Framework ◽  
The Self ◽  
Open Set Condition ◽  
Open Set ◽  
Self Similar ◽  
Bernoulli Measures ◽  
Multifractal Measures

Abstract M. Das proved that the relative multifractal measures are mutually singular for the self-similar measures satisfying the significantly weaker open set condition. The aim of this paper is to show that these measures are mutually singular in a more general framework. As examples, we apply our main results to quasi-Bernoulli measures.

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