ON THE ASSOUAD DIMENSION OF GRAPH DIRECTED MORAN FRACTALS

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.

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The quasi-Assouad dimension of stochastically self-similar sets

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.112 ◽

2019 ◽

Vol 150 (1) ◽

pp. 261-275 ◽

Cited By ~ 2

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.

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Lower Assouad Dimension of Measures and Regularity

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004119000458 ◽

2019 ◽

pp. 1-37 ◽

Cited By ~ 1

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.

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The Hausdorff dimension and exact Hausdorff measure of random recursive sets with overlapping

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202110337 ◽

2002 ◽

Vol 31 (1) ◽

pp. 11-21 ◽

Cited By ~ 1

Author(s):

Hongwen Guo ◽

Dihe Hu

Keyword(s):

Hausdorff Dimension ◽

Hausdorff Measure ◽

Intersection Property ◽

Random Sets ◽

Hausdorff Measures ◽

Open Set Condition ◽

Finite Intersection ◽

Open Set ◽

Finite Intersection Property

We weaken the open set condition and define a finite intersection property in the construction of the random recursive sets. We prove that this larger class of random sets are fractals in the sense of Taylor, and give conditions when these sets have positive and finite Hausdorff measures, which in certain extent generalize some of the known results, about random recursive fractals.

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Open set condition for graph directed self-similar structure

Mathematische Zeitschrift ◽

10.1007/s00209-013-1196-z ◽

2013 ◽

Vol 276 (1-2) ◽

pp. 243-260 ◽

Cited By ~ 2

Author(s):

Tian-jia Ni ◽

Zhi-ying Wen

Keyword(s):

Open Set Condition ◽

Open Set ◽

Self Similar

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

Fractals ◽

10.1142/s0218348x20500280 ◽

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.

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

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.26 ◽

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.

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