Bi-H"older invariants in o-minimal structures

We prove that for any two definable germs in a polynomially bounded o-minimal structure, there exists a critical threshold $\alpha_0 \in (0,1)$ such that if these germs are bi-$\alpha$-H"older equivalent for some $\alpha \ge \alpha_0$, then they satisfy the following: \begin{itemize}[label=$\circ$] \item The Lipschitz normal embedding (LNE) property is preserved; that is, if one germ is LNE then so is the other; \item Their tangent cones have the same dimension; \item The links of their tangent cones have isomorphic homotopy groups. \end{itemize} As an application, we give an simple proof that a complex analytic germ which is bi-$\alpha$-H"older homeomorphic to the germ of a Euclidean space for some $\alpha$ sufficiently close to $1$ must be smooth. This provides a slightly stronger version of Sampaio's smoothness theorem, in which the germs are assumed to be bi-$\alpha$-H"older homeomorphic for every $\alpha \in (0,1)$.