We present a complete bi-Lipschitz classification of germs of semialgebraic curves (semialgebraic sets of the dimension one). For this purpose we introduce the so-called Hölder Semicomplex, a bi-Lipschitz invariant. Hölder Semicomplex is the collection of all first exponents of Newton-Puiseux expansions, for all pairs of branches of a curve. We prove that two germs of curves are bi-Lipschitz equivalent if and only if the corresponding Hölder Semicomplexes are isomorphic. We also prove that any Hölder Semicomplex can be realized as a germ of some plane semialgebraic curve. Finally, we compare these Hölder Semicomplexes with Hölder Complexes-complete bi-Lipschitz invariant of two-dimensional semialgebraic sets.
@article{urn:eudml:doc:44492,
title = {Teor\'\i a m\'etrica de curvas semialgebr\'aicas.},
journal = {Revista Matem\'atica de la Universidad Complutense de Madrid},
volume = {13},
year = {2000},
pages = {369-382},
zbl = {0979.14027},
mrnumber = {MR1822120},
language = {es},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44492}
}
Birbrair, Lev; Fernandes, Alexandre C. G. Teoría métrica de curvas semialgebráicas.. Revista Matemática de la Universidad Complutense de Madrid, Tome 13 (2000) pp. 369-382. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44492/