How inhomogeneous Cantor sets can pass a point

For x > 0 , let Υ ( x ) = ( a , b ) : x ∈ E a , b , a > 0 , b > 0 , a + b ≤ 1 , where E a , b is the unique nonempty compact invariant set generated by the inhomogeneous IFS Ψ a , b = f 0 ( x ) = a x , f 1 ( x ) = b ( x + 1 ) . We show that the set Υ ( x ) is a Lebesgue null set with full H...

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Bibliographic Details
Published inMathematische Zeitschrift Vol. 302; no. 3; pp. 1429 - 1449
Main Authors Li, Wenxia, Wang, Zhiqiang
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2022
Springer Nature B.V
Subjects
Mathematics
Mathematics and Statistics
Thickness
Intersection
Hausdorff dimension
Cantor set
Primary 28A78
Secondary 11A63
37B10
Inhomogeneous
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Summary:For x > 0 , let Υ ( x ) = ( a , b ) : x ∈ E a , b , a > 0 , b > 0 , a + b ≤ 1 , where E a , b is the unique nonempty compact invariant set generated by the inhomogeneous IFS Ψ a , b = f 0 ( x ) = a x , f 1 ( x ) = b ( x + 1 ) . We show that the set Υ ( x ) is a Lebesgue null set with full Hausdorff dimension and the intersection of the sets Υ ( x 1 ) , … , Υ ( x ℓ ) still has full Hausdorff dimension for any finite number of positive numbers x 1 , … , x ℓ .
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-022-03099-0