zbMATH — the first resource for mathematics

Invariants of bi-Lipschitz equivalence of real analytic functions. (English) Zbl 1059.32006
Hironaka, Heisuke (ed.) et al., Geometric singularity theory. Dedicated to the memory of Stanisław Łojasiewicz. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 65, 67-75 (2004).
An invariant of the bi-Lipschitz equivalence of analytic function germs \(f:(\mathbb R^n,0) \rightarrow (\mathbb R,0)\) is constructed which varies continuously in many analytic families. The invariant is given in terms of the leading coefficients of the asymptotic expansions of \(f\) along the sets where the size of \(| x| | \text{grad} (f(x))| \) is comparabel to the size of \(| f(x)| \). As an example let \(f_t(x,y) = x^3 - 3t x y^4 + y^6\) define a one parameter family of germs \(f_t : (\mathbb R^2,0) \to (\mathbb R,0)\), \(t \in \mathbb R\). It is proved that if \(t,t^\prime\) are sufficiently generic, then \(f_t\) and \(f_{t^\prime}\) are not bi-Lipschitz equivalent function germs, that is, there is no germ of bi-Lipschitz homeomorphism \(H : (\mathbb R^2,0) \to (\mathbb R^2,0)\) such that \(f_t \circ H = f_{t^\prime}\).
For the entire collection see [Zbl 1051.57001].

32S15 Equisingularity (topological and analytic)
32S05 Local complex singularities
14H15 Families, moduli of curves (analytic)