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

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

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

Let A,B n be subanalytic set-germs at 0 n such that 0A ¯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.

DOI : 10.5802/aif.2496
Classification : 14P15, 32B20, 57R45
Keywords: Subanalytic set, direction set, bi-Lipschitz homeomorphism
Mot 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)
@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/

[1] Bierstone, E.; Milman, P. D. Arc-analytic functions, Invent. math., Volume 101 (1990), pp. 411-424 | DOI | MR | Zbl

[2] Bochnak, J.; Risler, J.-J. Sur les exposants de Lojasiewicz, Comment. Math. Helv., Volume 50 (1975), pp. 493-507 | DOI | MR | Zbl

[3] Briançon, J.; Speder, J. P. La trivialité topologique n’implique pas les conditions de Whitney, C. R. Acad. Sci. Paris, Volume 280 (1975), pp. 365-367 | MR | Zbl

[4] Fukui, T. The modified analytic trivialization via the weighted blowing up, J. Math. Soc. Japan, Volume 44 (1992), pp. 455-459 | DOI | MR | Zbl

[5] Fukui, T.; Koike, S.; Kuo, T.-C.; T. Fukuda, T. Fukui, S. Izumiya and S. Koike Blow-analytic equisingularities, properties, problems and progress, Real Analytic and Algebraic Singularities (Pitman Research Notes in Mathematics Series), Volume 381, Longman, 1998, pp. 8-29 | MR | Zbl

[6] Fukui, T.; Paunescu, L. Modified analytic trivialization for weighted homogeneous function-germs, J. Math. Soc. Japan, Volume 52 (2000), pp. 433-446 | DOI | MR | Zbl

[7] Fukui, T.; Paunescu, L.; Coste, M.; Kurdyka, K.; McCrory, C.; Parusinski, A. Arc Spaces and additive invariants in real algebraic and analytic geometry, Panoramas et Synthèses, Société Mathématique de France, 2008 no. 24 | MR | Zbl

[8] Henry, J.-P.; Parusiński, A. Existence of Moduli for bi-Lipschitz equivalence of analytic functions, Compositio Math., Volume 136 (2003), pp. 217-235 | DOI | MR | Zbl

[9] Henry, J.-P.; Parusiński, A. Invariants of bi-Lipschitz equivalence of real analytic functions, Banach Center Publications, Volume 65 (2004), pp. 67-75 | DOI | MR | Zbl

[10] Hironaka, H. Subanalytic sets, Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 453-493 | MR | Zbl

[11] Hironaka, H.; Oslo 1976, P. Holm Stratification and flatness, Real and Complex Singularities, Sithoff and Noordhoff, 1977, pp. 196-265 | MR | Zbl

[12] Koike, S. On strong C 0 -equivalence of real analytic functions, J. Math. Soc. Japan, Volume 45 (1993), pp. 313-320 | DOI | MR | Zbl

[13] Koike, S. The Briançon-Speder and Oka families are not biLipschitz trivial, Several Topics in Singularity Theory, RIMS Kokyuroku, Volume 1328 (2003), pp. 165-173 | Zbl

[14] Kuo, T.-C. A complete determination of C 0 -sufficiency in J r (2,1), Invent. math., Volume 8 (1969), pp. 226-235 | DOI | MR | Zbl

[15] Kuo, T.-C. Characterizations of v-sufficiency of jets, Topology, Volume 11 (1972), pp. 115-131 | DOI | MR | Zbl

[16] Kuo, T.-C. Une classification des singularités réels, C.R. Acad. Sci. Paris, Volume 288 (1979), pp. 809-812 | MR | Zbl

[17] Kuo, T.-C. The modified analytic trivialization of singularities, J. Math. Soc. Japan , Volume 32 (1980), pp. 605-614 | DOI | MR | Zbl

[18] Kuo, T.-C. On classification of real singularities, Invent. math., Volume 82 (1985), pp. 257-262 | DOI | MR | Zbl

[19] Kurdyka, K. Ensembles semi-algébriques symétriques par arcs, Math. Ann., Volume 282 (1988), pp. 445-462 | DOI | MR | Zbl

[20] Lojasiewicz, S. Ensembles semi-analytiques, Inst. Hautes Etudes Sci. Lectute Note (1967)

[21] Mostowski, T. Lipschitz equisingularity, 243, Dissertationes Math., 1985 | MR | Zbl

[22] Mostowski, T. A criterion for Lipschitz equisingularity, Bull. Acad. Polon. Sci., Volume 37 (1988), pp. 109-116 | MR | Zbl

[23] Mostowski, T. Lipschitz equisingularity problems, Several Topics in Singularity Theory, RIMS Kokyuroku, Volume 1328 (2003), pp. 73-113 | Zbl

[24] Oka, M. On the weak simultaneous resolution of a negligible truncation of the Newton boundary, Contemporary Math., Volume 90 (1989), pp. 199-210 | MR | Zbl

[25] Parusiński, A. Lipschitz properties of semi-analytic sets, Ann. Inst. Fourier, Volume 38 (1988), pp. 189-213 | DOI | Numdam | MR | Zbl

[26] Parusiński, A. Lipschitz stratification of real analytic sets, Singularities, Banach Center Publications, Volume 20 (1988), pp. 323-333 | MR | Zbl

[27] Parusiński, A. Lipschitz stratification of subanalytic sets, Ann. Sci. Ec. Norm. Sup., Volume 27 (1994), pp. 661-696 | Numdam | MR | Zbl

[28] Paunescu, L.; T. Fukuda, T. Fukui, S. Izumiya and S. Koike An example of blow-analytic homeomorphism, Real Analytic and Algebraic Singularities (Pitman Research Notes in Mathematics Series), Volume 381, Longman, 1998, pp. 62-63 | MR | Zbl

Cité par Sources :