no. 6
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, Satoshi1 ; Paunescu, Laurentiu2
1 Hyogo University of Teacher Education Department of Mathematics Kato, Hyogo 673-1494 (Japan)
2 University of Sydney School of Mathematics and Statistics Sydney, NSW, 2006 (Australia)
Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2445-2467.

Let A⊂ℝ n be a set-germ at 0∈ℝ n such that 0∈A ¯. We say that r∈S n-1 is a direction of A at 0∈ℝ n if there is a sequence of points {x i }⊂A∖{0} tending to 0∈ℝ n such that x i ∥x i ∥→r as i→∞. Let D(A) denote the set of all directions of A at 0∈ℝ n .

Let A,B⊂ℝ n be subanalytic set-germs at 0∈ℝ n such that 0∈A ¯∩B ¯. We study the problem of whether the dimension of the common direction set, dim(D(A)∩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.

Soit A⊂ℝ n un germe d’ensemble en 0∈ℝ n tel que 0∈A ¯. On dit que r∈S n-1 est une direction de A en 0∈ℝ n s’il existe une suite de points {x i }⊂A∖{0} qui converge vers 0∈ℝ n telle que x i ∥x i ∥→r quand i→∞. L’ensemble des directions de A en 0∈ℝ n est noté D(A). Soient A,B⊂ℝ n deux germes en 0∈ℝ n d’ensemble sous-analytique tels que 0∈A ¯∩B ¯.

On étudie le problème suivant : la dimension de l’intersection, dim(D(A)∩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.

DOI : 10.5802/aif.2496
Classification : 14P15, 32B20, 57R45
Keywords: Subanalytic set, direction set, bi-Lipschitz homeomorphism
Mots-clés : ensemble sous-analytique, dimension de l’intersection, homéomorphisme bi-Lipschitzien

Koike, Satoshi 1 ; Paunescu, Laurentiu 2

1 Hyogo University of Teacher Education Department of Mathematics Kato, Hyogo 673-1494 (Japan)
2 University of Sydney School of Mathematics and Statistics Sydney, NSW, 2006 (Australia) 
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2496/

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