Full paper in PDF:
$%L. Birbrair and A. C. G. Fernandes, Metric theory of semialgebraic curves, Rev. Mat. Complut. 13 (2000), no. 2, 369–382.%$

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.

1991 Mathematics Subject Classification: 14P25.