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

Koike, Satoshi; Paunescu, Laurentiu

Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2445-2467.

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

On étudie le problème suivant : la dimension de l’intersection, $dim\left(D\right(A)\cap D(B\left)\right)$, 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.

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

Let $A,B\subset {ℝ}^n$ be subanalytic set-germs at $0\in {ℝ}^n$ such that $0\in \overline{A}\cap \overline{B}$. We study the problem of whether the dimension of the common direction set, $dim\left(D\right(A)\cap D(B\left)\right)$ 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.

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