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 -