Suppose I have a smooth Riemannian manifold $X$ with induced distance function $d$, and a bi-Lipschitz (with respect to $d$) homeomorphism
$$\phi: X \to X.$$


Under what circumstances could $\phi$ be smoothable to a diffeomorphism? By "smoothable" in this case I mean "homotopic to a diffeomorphism through bi-Lipschitz homeomorphisms" (this might not be standard, I suppose). Are there any clear obstructions?



When is a bi-Lipschitz homeomorphism smoothable?
Any self-homeomorphism of a manifold of dimension $\neq 4$ is topologically isotopic to a bi-Lipschitz homeomorphism, see lemma 2.4 in [Lipschitz and quasiconformal approximation of homeomorphism pairs](https://www.sciencedirect.com/science/article/pii/S0166864199001455)
by Jouni Luukkainen.


There are exotic spheres (e.g. in dimension $7$) that admit a self-homeomorphism that is not homotopic to a diffeomorphism, see [here](https://mathoverflow.net/a/348603/1573). This gives a bi-Lipschitz homeomorphism that is not homotopic to a diffeomorphism.


