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.
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.
Accepted:
DOI: 10.5802/aif.2496
Classification: 14P15, 32B20, 57R45
Keywords: Subanalytic set, direction set, bi-Lipschitz homeomorphism
@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 = {https://aif.centre-mersenne.org/articles/10.5802/aif.2496/} }
TY - JOUR TI - The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms JO - Annales de l'Institut Fourier PY - 2009 DA - 2009/// SP - 2445 EP - 2467 VL - 59 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2496/ UR - https://zbmath.org/?q=an%3A1184.14086 UR - https://www.ams.org/mathscinet-getitem?mr=2640926 UR - https://doi.org/10.5802/aif.2496 DO - 10.5802/aif.2496 LA - en ID - AIF_2009__59_6_2445_0 ER -
%0 Journal Article %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 https://doi.org/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, Volume 59 (2009) no. 6, pp. 2445-2467. doi : 10.5802/aif.2496. https://aif.centre-mersenne.org/articles/10.5802/aif.2496/
[1] Arc-analytic functions, Invent. math., Tome 101 (1990), pp. 411-424 | Article | MR: 1062969 | Zbl: 0723.32005
[2] Sur les exposants de Lojasiewicz, Comment. Math. Helv., Tome 50 (1975), pp. 493-507 | Article | MR: 404674 | Zbl: 0321.32006
[3] La trivialité topologique n’implique pas les conditions de Whitney, C. R. Acad. Sci. Paris, Tome 280 (1975), pp. 365-367 | MR: 425165 | Zbl: 0331.32010
[4] The modified analytic trivialization via the weighted blowing up, J. Math. Soc. Japan, Tome 44 (1992), pp. 455-459 | Article | MR: 1167377 | Zbl: 0766.58008
[5] Blow-analytic equisingularities, properties, problems and progress, Real Analytic and Algebraic Singularities (T. Fukuda, T. Fukui, S. Izumiya and S. Koike, ed.) (Pitman Research Notes in Mathematics Series) Tome 381, Longman, 1998, pp. 8-29 | MR: 1607662 | Zbl: 0954.26012
[6] Modified analytic trivialization for weighted homogeneous function-germs, J. Math. Soc. Japan, Tome 52 (2000), pp. 433-446 | Article | MR: 1742795 | Zbl: 0964.32023
[7] Arc Spaces and additive invariants in real algebraic and analytic geometry, Panoramas et Synthèses, Société Mathématique de France, 2008 no. 24 | MR: 2404096 | Zbl: 1155.14313
[8] Existence of Moduli for bi-Lipschitz equivalence of analytic functions, Compositio Math., Tome 136 (2003), pp. 217-235 | Article | MR: 1967391 | Zbl: 1026.32055
[9] Invariants of bi-Lipschitz equivalence of real analytic functions, Banach Center Publications, Tome 65 (2004), pp. 67-75 | Article | MR: 2104338 | Zbl: 1059.32006
[10] Subanalytic sets, Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 453-493 | MR: 377101 | Zbl: 0297.32008
[11] Stratification and flatness, Real and Complex Singularities (Oslo 1976, P. Holm, ed.), Sithoff and Noordhoff, 1977, pp. 196-265 | MR: 499286 | Zbl: 0424.32004
[12] On strong -equivalence of real analytic functions, J. Math. Soc. Japan, Tome 45 (1993), pp. 313-320 | Article | MR: 1206656 | Zbl: 0788.32024
[13] The Briançon-Speder and Oka families are not biLipschitz trivial, Several Topics in Singularity Theory, RIMS Kokyuroku, Tome 1328 (2003), pp. 165-173 | Zbl: 1064.58031
[14] A complete determination of -sufficiency in , Invent. math., Tome 8 (1969), pp. 226-235 | Article | MR: 254860 | Zbl: 0183.04602
[15] Characterizations of -sufficiency of jets, Topology, Tome 11 (1972), pp. 115-131 | Article | MR: 288775 | Zbl: 0234.58005
[16] Une classification des singularités réels, C.R. Acad. Sci. Paris, Tome 288 (1979), pp. 809-812 | MR: 535641 | Zbl: 0404.58013
[17] The modified analytic trivialization of singularities, J. Math. Soc. Japan , Tome 32 (1980), pp. 605-614 | Article | MR: 589100 | Zbl: 0509.58007
[18] On classification of real singularities, Invent. math., Tome 82 (1985), pp. 257-262 | Article | MR: 809714 | Zbl: 0587.32018
[19] Ensembles semi-algébriques symétriques par arcs, Math. Ann., Tome 282 (1988), pp. 445-462 | Article | MR: 967023 | Zbl: 0686.14027
[20] Ensembles semi-analytiques, Inst. Hautes Etudes Sci. Lectute Note (1967)
[21] Lipschitz equisingularity Tome 243, Dissertationes Math., 1985 | MR: 808226 | Zbl: 0578.32020
[22] A criterion for Lipschitz equisingularity, Bull. Acad. Polon. Sci., Tome 37 (1988), pp. 109-116 | MR: 1101458 | Zbl: 0761.32018
[23] Lipschitz equisingularity problems, Several Topics in Singularity Theory, RIMS Kokyuroku, Tome 1328 (2003), pp. 73-113 | Zbl: 1064.58032
[24] On the weak simultaneous resolution of a negligible truncation of the Newton boundary, Contemporary Math., Tome 90 (1989), pp. 199-210 | MR: 1000603 | Zbl: 0682.32011
[25] Lipschitz properties of semi-analytic sets, Ann. Inst. Fourier, Tome 38 (1988), pp. 189-213 | Article | Numdam | MR: 978246 | Zbl: 0631.32006
[26] Lipschitz stratification of real analytic sets, Singularities, Banach Center Publications, Tome 20 (1988), pp. 323-333 | MR: 1101849 | Zbl: 0666.32011
[27] Lipschitz stratification of subanalytic sets, Ann. Sci. Ec. Norm. Sup., Tome 27 (1994), pp. 661-696 | Numdam | MR: 1307677 | Zbl: 0819.32007
[28] An example of blow-analytic homeomorphism, Real Analytic and Algebraic Singularities (T. Fukuda, T. Fukui, S. Izumiya and S. Koike, ed.) (Pitman Research Notes in Mathematics Series) Tome 381, Longman, 1998, p. 62-63 | MR: 1607678 | Zbl: 0896.58012
Cited by Sources: