On the Invariance of the Real Milnor Number under Asymptotically Lipschitz Equivalence

We investigate sufficient conditions for the invariance of the real Milnor number under $\mathcal{R}$-bi-Lipschitz equivalence for function-germs $f,g\text{\hspace{0.17em}⁣}:({\mathbb{R}}^n,0)\to (\mathbb{R},0)$. 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 $C^{\mathrm{\infty }}$ equivalences, establishing sufficient conditions for the real Milnor number to remain invariant.