On a kind of self-similar sets with complete overlaps

Kong, Derong; Yao, Yi Yong

doi:10.1007/s10474-020-01116-4

Cited by 2 publications

(1 citation statement)

References 19 publications

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“…Lau and Ngai [43] defined the WSC in slightly different form to analyse the multifractal structure of one-dimensional overlapping self-similar measures, following Lalley [42]. One-dimensional WSC attractors and measures have been further investigated by Feng [27,28], Hare and Hare with coauthors [35,36,37] and others [21,40]. There are few studies of two-dimensional WSC examples [12,19,64], and apparently no studies for greater dimension.…”

Section: Weak Separationmentioning

confidence: 99%

Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

Bandt

Mekhontsev

2022

Fractal Fract

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Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields.

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Section: Weak Separationmentioning

confidence: 99%

Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

Bandt

Mekhontsev

2022

Fractal Fract

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Elementary Fractal Geometry. 3. Complex Pisot Factors Imply Finite Type

Bandt

2024

Discrete Comput Geom

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Self-similar sets require a separation condition to admit a nice mathematical structure. The classical open set condition (OSC) is difficult to verify. Zerner proved that there is a positive and finite Hausdorff measure for a weaker separation property which is always fulfilled for crystallographic data. Ngai and Wang gave more specific results for a finite type property (FT), and for algebraic data with a real Pisot expansion factor. We show how the algorithmic FT concept of Bandt and Mesing relates to the property of Ngai and Wang. Merits and limitations of the FT algorithm are discussed. Our main result says that FT is always true in the complex plane if the similarity mappings are given by a complex Pisot expansion factor $$\lambda $$ λ and algebraic integers in the number field generated by $$\lambda .$$ λ . This extends the previous results and opens the door to huge classes of separated self-similar sets, with large complexity and an appearance of natural textures. Numerous examples are provided.

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On a kind of self-similar sets with complete overlaps

Cited by 2 publications

References 19 publications

Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

Elementary Fractal Geometry. 3. Complex Pisot Factors Imply Finite Type

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