Affine embeddings and intersections of Cantor sets

2014/06/20 by Feng, De-Jun, Huang, Wen, Rao, Hui · 1 citation
#28A75 #37C45 #37D35 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1406.5318

Abstract

Let E, F⊂ \R^d be two self-similar sets. Under mild conditions, we show that F can be C¹-embedded into E if and only if it can be affinely embedded into E; furthermore if F can not be affinely embedded into E, then the Hausdorff dimension of the intersection E∩ f(F) is strictly less than that of F for any C¹-diffeomorphism f on \R^d. Under certain circumstances, we prove the logarithmic commensurability between the contraction ratios of E and F if F can be affinely embedded into E. As an application, we show that dim_HE∩ f(F)

Cited by

• From Lipschitz embedding to Lipschitz equivalence between dust-like self-similar sets

Related

• Affine embeddings of Cantor sets and dimension of αβ-sets
• Affine embeddings of Cantor sets on the line
• Affine embeddings of Cantor sets in the plane
• Dimension drop for intersections of Cantor sets
• On Intersections of Cantor Sets: Self-Similarity