We investigate sufficient conditions for the invariance of the real Milnor number under $\mathcal{ R}$-bi-Lipschitz equivalence for function-germs $ f, g \colon (\mathbb{ R}^n, 0) \to (\mathbb{ R}, 0) $. More generally, we explore its invariance within the extended framework of $\mathcal{ R}$-asymptotically Lipschitz equivalence. To this end, we introduce the $\alpha$-derivative, which provides a natural setting for studying asymptotic growth. Additionally, we discuss the implications of our results in the context of $C^k$ and $C^{ \infty}$ equivalences, establishing sufficient conditions for the real Milnor number to remain invariant.
Categories: math.AG math.MG