2012/02/20 by Jean Bellissard, Jean V. Bellissard, Bellissard, Jean V. +2
Computer Science · Mathematics · #30L05 (primary) 37B10 #37B50 (secondary) #60J80 #FOS: Mathematics #General Topology (math.GN) #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #math.GN #msc:30L05 #msc:37B10 #msc:37B50 #msc:60J80
paper · pdf · doi:10.48550/arxiv.1202.4330
30 pages, 1 figure
openalex publication_date 2012/02/20 · arxiv created 2013/10/22 · arxiv updated 2013/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
An ultrametric Cantor set can be seen as the boundary of a rooted weighted tree called the Michon tree. The notion of Assouad dimension is re-interpreted as seen on the Michon tree. The Assouad dimension of an ultrametric Cantor set is finite if and only if the space is bi-Lipschitz embeddable in a finite dimensional Euclidean space. This result, due to Assouad and refined by Luukkainen--Movahedi-Lankarani is re-proved in the Michon tree formalism. It is applied to answer the embedding question for some spaces which can be seen naturally as boundary of trees: linearly repetitive subshifts, Sturmian subshifts, and the boundary of Galton--Watson trees with random weights. Some of these give examples of nonembeddable spaces with finite Hausdorff dimension.