The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms

[La dimension directionnelle des ensembles sous-analytiques est invariante par les homéomorphismes bi-Lipschitz]

Koike, Satoshi ; Paunescu, Laurentiu

Annales de l'Institut Fourier, Tome 59 (2009), p. 2445-2467 / Harvested from Numdam

Résumé

Soit $A \subset \mathbb{R}^n$ un germe d’ensemble en $0 \in \mathbb{R}^n$ tel que $0 \in \overline{A}$. On dit que $r \in S^{n-1}$ est une direction de $A$ en $0 \in \mathbb{R}^n$ s’il existe une suite de points $\lbrace x_i \rbrace \subset A \setminus \lbrace 0 \rbrace $ qui converge vers $0 \in \mathbb{R}^n$ telle que ${x_i \over \Vert x_i \Vert } \rightarrow r$ quand $i \rightarrow \infty $. L’ensemble des directions de $A$ en $0 \in \mathbb{R}^n$ est noté $D(A)$. Soient $A, \ B \subset \mathbb{R}^n$ deux germes en $0 \in \mathbb{R}^n$ d’ensemble sous-analytique tels que $0 \in \overline{A} \cap \overline{B}$.

On étudie le problème suivant : la dimension de l’intersection, $\dim (D(A) \cap D(B))$, est-elle invariante par homéomorphisme bi-Lipschitzien ? On montre que la réponse est non en général, néanmoins la propriété est vraie, lorsque les images de $A$ et $B$ sont sous-analytiques. En particulier, les ensembles des directions de deux germes sous-analytiques, équivalents par homéomorphisme bi-Lipschitzien, ont la même dimension.

Abstract

Let $A \subset \mathbb{R}^n$ be a set-germ at $0 \in \mathbb{R}^n$ such that $0 \in \overline{A}$. We say that $r \in S^{n-1}$ is a direction of $A$ at $0 \in \mathbb{R}^n$ if there is a sequence of points $\lbrace x_i \rbrace \subset A \setminus \lbrace 0 \rbrace $ tending to $0 \in \mathbb{R}^n$ such that ${x_i \over \Vert x_i \Vert } \rightarrow r$ as $i \rightarrow \infty $. Let $D(A)$ denote the set of all directions of $A$ at $0 \in \mathbb{R}^n$.

Let $A, \ B \subset \mathbb{R}^n$ be subanalytic set-germs at $0 \in \mathbb{R}^n$ such that $0 \in \overline{A} \cap \overline{B}$. We study the problem of whether the dimension of the common direction set, $\dim (D(A) \cap D(B))$ is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of $A$ and $B$ are also subanalytic. In particular if two subanalytic set-germs are bi-Lipschitz equivalent their direction sets must have the same dimension.

Détail

Publié le : 2009-01-01

MR 2640926 | Zbl 1184.14086 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2496
Classification:  14P15,  32B20,  57R45
Mots clés: ensemble sous-analytique, dimension de l’intersection, homéomorphisme bi-Lipschitzien

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