Mathematics – General Topology
Scientific paper
2012-02-20
Mathematics
General Topology
31 pages, 2 figures
Scientific paper
Let $(C,d)$ be an ultrametric Cantor set. Then it admits an isometric embedding into an infinite dimensional Euclidean space \cite{PB08}. Associated with it is a weighted rooted tree, the reduced Michon graph $\mathscr T$ \cite{Mi85}. It will be called $f$-embeddable if there is a bi-Lipshitz map from $(C,d)$ into a finite dimensional Euclidean space. The main result establishes that $(C,d)$ is $f$-embeddable if and only if it can be represented by a weighted Michon tree such that (i) the number of children per vertex is uniformly bounded, (ii) if $\kappa$ denotes the weight, there are constants $c>0$ and $0<\delta<1$ such that $\kappa(v)/\kappa(u)\leq c\,\delta^{d(u,v)}$ where $v$ is a descendant of $u$ and where $d(u,v)$ denotes the graph distance between the vertices $u,v$. Several examples are provided: (a) the tiling space of a linear repetitive sequence is $f$-embeddable, (b) the tiling space of a Sturmian sequence is $f$-embeddable if and only if the irrational number characterizing it has bounded type, (c) the boundary of a Galton-Watson random trees with more than one child per vertex is almost surely not $f$-embeddable.
Bellissard Jean V.
Julien Antoine
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