On one-point intersection property for self-similar fractals

Kirill Kamalutdinov; Andrei Tetenov

doi:10.1088/1361-6544/ab4e0e

On one-point intersection property for self-similar fractals

Research output: Contribution to journal › Article

Abstract

This was a long-standing question since 90s whether one-point intersection property for a self-similar set implies open set condition (OSC). We answer this question negatively. We give an example of a totally disconnected self-similar set K ⊂ ℝ which does not have OSC and has minimal overlap of its pieces, that is, all intersections of its pieces K_i∩K_j, i≠j are empty except only one, which is a single point.

Original language	English
Pages (from-to)	408-416
Number of pages	9
Journal	Nonlinearity
Volume	33
Issue number	1
DOIs	https://doi.org/10.1088/1361-6544/ab4e0e
Publication status	Published - 1 Jan 2020

Keywords

general position theorem
Hausdorff dimension
open set condition
self-similar set
weak separation property

Access to Document

10.1088/1361-6544/ab4e0e

Link to publication in Scopus

Fingerprint Dive into the research topics of 'On one-point intersection property for self-similar fractals'. Together they form a unique fingerprint.

Cite this

Kamalutdinov, K., & Tetenov, A. (2020). On one-point intersection property for self-similar fractals. Nonlinearity, 33(1), 408-416. https://doi.org/10.1088/1361-6544/ab4e0e

@article{d6c627e9d9e64ebe87ad214bcd178888,

title = "On one-point intersection property for self-similar fractals",

abstract = "This was a long-standing question since 90s whether one-point intersection property for a self-similar set implies open set condition (OSC). We answer this question negatively. We give an example of a totally disconnected self-similar set K ⊂ ℝ which does not have OSC and has minimal overlap of its pieces, that is, all intersections of its pieces Ki∩Kj, i≠j are empty except only one, which is a single point.",

keywords = "general position theorem, Hausdorff dimension, open set condition, self-similar set, weak separation property",

author = "Kirill Kamalutdinov and Andrei Tetenov",

year = "2020",

month = jan,

day = "1",

doi = "10.1088/1361-6544/ab4e0e",

language = "English",

volume = "33",

pages = "408--416",

journal = "Nonlinearity",

issn = "0951-7715",

publisher = "IOP Publishing Ltd.",

number = "1",

}

TY - JOUR

T1 - On one-point intersection property for self-similar fractals

AU - Kamalutdinov, Kirill

AU - Tetenov, Andrei

PY - 2020/1/1

Y1 - 2020/1/1

N2 - This was a long-standing question since 90s whether one-point intersection property for a self-similar set implies open set condition (OSC). We answer this question negatively. We give an example of a totally disconnected self-similar set K ⊂ ℝ which does not have OSC and has minimal overlap of its pieces, that is, all intersections of its pieces Ki∩Kj, i≠j are empty except only one, which is a single point.

AB - This was a long-standing question since 90s whether one-point intersection property for a self-similar set implies open set condition (OSC). We answer this question negatively. We give an example of a totally disconnected self-similar set K ⊂ ℝ which does not have OSC and has minimal overlap of its pieces, that is, all intersections of its pieces Ki∩Kj, i≠j are empty except only one, which is a single point.

KW - general position theorem

KW - Hausdorff dimension

KW - open set condition

KW - self-similar set

KW - weak separation property

UR - http://www.scopus.com/inward/record.url?scp=85081312195&partnerID=8YFLogxK

U2 - 10.1088/1361-6544/ab4e0e

DO - 10.1088/1361-6544/ab4e0e

M3 - Article

AN - SCOPUS:85081312195

VL - 33

SP - 408

EP - 416

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 1

ER -