Soit un germe d’ensemble en tel que . On dit que est une direction de en s’il existe une suite de points qui converge vers telle que quand . L’ensemble des directions de en est noté . Soient deux germes en d’ensemble sous-analytique tels que .
On étudie le problème suivant : la dimension de l’intersection, , 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 et sont sous-analytiques. En particulier, les ensembles des directions de deux germes sous-analytiques, équivalents par homéomorphisme bi-Lipschitzien, ont la même dimension.
Let be a set-germ at such that . We say that is a direction of at if there is a sequence of points tending to such that as . Let denote the set of all directions of at .
Let be subanalytic set-germs at such that . We study the problem of whether the dimension of the common direction set, is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of and are also subanalytic. In particular if two subanalytic set-germs are bi-Lipschitz equivalent their direction sets must have the same dimension.
Keywords: Subanalytic set, direction set, bi-Lipschitz homeomorphism
Mot clés : ensemble sous-analytique, dimension de l’intersection, homéomorphisme bi-Lipschitzien
@article{AIF_2009__59_6_2445_0, author = {Koike, Satoshi and Paunescu, Laurentiu}, title = {The directional dimension of subanalytic sets is invariant under {bi-Lipschitz} homeomorphisms}, journal = {Annales de l'Institut Fourier}, pages = {2445--2467}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {6}, year = {2009}, doi = {10.5802/aif.2496}, zbl = {1184.14086}, mrnumber = {2640926}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2496/} }
TY - JOUR AU - Koike, Satoshi AU - Paunescu, Laurentiu TI - The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms JO - Annales de l'Institut Fourier PY - 2009 SP - 2445 EP - 2467 VL - 59 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2496/ DO - 10.5802/aif.2496 LA - en ID - AIF_2009__59_6_2445_0 ER -
%0 Journal Article %A Koike, Satoshi %A Paunescu, Laurentiu %T The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms %J Annales de l'Institut Fourier %D 2009 %P 2445-2467 %V 59 %N 6 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2496/ %R 10.5802/aif.2496 %G en %F AIF_2009__59_6_2445_0
Koike, Satoshi; Paunescu, Laurentiu. The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms. Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2445-2467. doi : 10.5802/aif.2496. http://www.numdam.org/articles/10.5802/aif.2496/
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