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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].

##### MSC:
 32S15 Equisingularity (topological and analytic) 32S05 Local complex singularities 14H15 Families, moduli of curves (analytic)
##### Keywords:
real analytic function; bi-Lipschitz equivalence