We construct an invariant of the bi-Lipschitz equivalence of analytic function germs (ℝⁿ,0) → (ℝ,0) that varies continuously in many analytic families. This shows that the bi-Lipschitz equivalence of analytic function germs admits continuous moduli. For a germ f the invariant is given in terms of the leading coefficients of the asymptotic expansions of f along the sets where the size of |x| |grad f(x)| is comparable to the size of |f(x)|.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc65-0-5,
author = {Jean-Pierre Henry and Adam Parusi\'nski},
title = {Invariants of bi-Lipschitz equivalence of real analytic functions},
journal = {Banach Center Publications},
volume = {65},
year = {2004},
pages = {67-75},
zbl = {1059.32006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc65-0-5}
}
Jean-Pierre Henry; Adam Parusiński. Invariants of bi-Lipschitz equivalence of real analytic functions. Banach Center Publications, Tome 65 (2004) pp. 67-75. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc65-0-5/