TY - JOUR
T1 - How likely can a point be in different Cantor sets
AU - Jiang, Kan
AU - Kong, Derong
AU - Li, Wenxia
N1 - Publisher Copyright:
© 2022 IOP Publishing Ltd & London Mathematical Society.
PY - 2022/3
Y1 - 2022/3
N2 - Given mN≥2, let K=Kλ:λ(0,1/m] be a class of self-similar sets with each Kλ='i=1∞diλi:di∈{0,1,⋯,m-1},i≥1 . In this paper we investigate the likelyhood of a point in the self-similar sets of K . More precisely, for a given point x ∈ (0, 1) we consider the parameter set Λ(x)=λ∈(0,1/m]:x∈Kλ, and show that Λ(x) is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, by constructing a sequence of Cantor subsets of Λ(x) with large thickness we show that for any x, y ∈ (0, 1) the intersection Λ(x) ∩ Λ(y) also has full Hausdorff dimension.
AB - Given mN≥2, let K=Kλ:λ(0,1/m] be a class of self-similar sets with each Kλ='i=1∞diλi:di∈{0,1,⋯,m-1},i≥1 . In this paper we investigate the likelyhood of a point in the self-similar sets of K . More precisely, for a given point x ∈ (0, 1) we consider the parameter set Λ(x)=λ∈(0,1/m]:x∈Kλ, and show that Λ(x) is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, by constructing a sequence of Cantor subsets of Λ(x) with large thickness we show that for any x, y ∈ (0, 1) the intersection Λ(x) ∩ Λ(y) also has full Hausdorff dimension.
KW - Hausdorff dimension
KW - intersection of Cantor sets
KW - self-similar set
KW - thickness of a Cantor set
UR - https://www.scopus.com/pages/publications/85125865837
U2 - 10.1088/1361-6544/ac4b3c
DO - 10.1088/1361-6544/ac4b3c
M3 - 文章
AN - SCOPUS:85125865837
SN - 0951-7715
VL - 35
SP - 1402
EP - 1430
JO - Nonlinearity
JF - Nonlinearity
IS - 3
ER -