Can I read Bi-Lipshitz Embedding of Ultrametric Cantor Sets into Euclidean Spaces on EtoBox?

Bi-Lipshitz Embedding of Ultrametric Cantor Sets into Euclidean Spaces by Bellissard, Jean V.; Julien, Antoine is a scholarly article available to read on EtoBox.

What is Bi-Lipshitz Embedding of Ultrametric Cantor Sets into Euclidean Spaces about?

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.

Author

Bellissard, Jean V.; Julien, Antoine

Published

2012

Language

EN