scholarly journals Assouad dimension of planar self-affine sets

2020 ◽  
Vol 374 (2) ◽  
pp. 1297-1326
Author(s):  
Balázs Bárány ◽  
Antti Käenmäki ◽  
Eino Rossi
Keyword(s):  
Assouad Dimension

scholarly journals Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

2020 ◽  
pp. 1-31
Author(s):  
Balázs Bárány ◽  
Károly Simon ◽  
István Kolossváry ◽  
Michał Rams
Keyword(s):  
Hausdorff Measure ◽  
Similar Case ◽  
Iterated Function Systems ◽  
The Self ◽  
Dimensional Hausdorff Measure ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Topological Sense

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

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.

scholarly journals The Assouad dimension of self-affine carpets with no grid structure

2017 ◽  
Vol 145 (11) ◽  
pp. 4905-4918 ◽  
Author(s):  
Jonathan M. Fraser ◽  
Thomas Jordan
Keyword(s):  
Grid Structure ◽  
Assouad Dimension

scholarly journals Badly approximable points on self-affine sponges and the lower Assouad dimension

2017 ◽  
Vol 39 (3) ◽  
pp. 638-657 ◽  
Author(s):  
TUSHAR DAS ◽  
LIOR FISHMAN ◽  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Assouad Dimension ◽  
Badly Approximable Points ◽  
Bounded Below ◽  
Badly Approximable ◽  
Sierpinski Sponges

We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. For self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpiński sponges/carpets (also known as Bedford–McMullen sponges/carpets) and the case of Barański carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.

Conformal assouad dimension and modulus

2004 ◽  
Vol 14 (6) ◽  
pp. 1278-1321 ◽  
Author(s):  
S. Keith ◽  
T. Laakso
Keyword(s):  
Assouad Dimension

scholarly journals LOWER ASSOUAD-TYPE DIMENSIONS OF UNIFORMLY PERFECT SETS IN DOUBLING METRIC SPACES

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050039
Author(s):  
HAIPENG CHEN ◽  
MIN WU ◽  
YUANYANG CHANG
Keyword(s):  
Metric Spaces ◽  
The Other ◽  
Equivalent Definition ◽  
Box Counting ◽  
Assouad Dimension ◽  
Uniformly Perfect ◽  
Box Counting Dimension ◽  
Definition Of ◽  
The Relationship ◽  
Variational Result

In this paper, we are concerned with the relationship among the lower Assouad-type dimensions. For uniformly perfect sets in doubling metric spaces, we obtain a variational result between two different but closely related lower Assouad spectra. As an application, we show that the limit of the lower Assouad spectrum as [Formula: see text] tends to 1 is equal to the quasi-lower Assouad dimension, which provides an equivalent definition to the latter. On the other hand, although the limit of the lower Assouad spectrum as [Formula: see text] tends to 0 exists, there exist uniformly perfect sets such that this limit is not equal to the lower box-counting dimension. Moreover, by the example of Cantor cut-out sets, we show that the new definition of quasi-lower Assouad dimension is more accessible, and indicate that the lower Assouad dimension could be strictly smaller than the lower spectra and the quasi-lower Assouad dimension.

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