Let $M$ and $N$ be smooth homeomorphic manifolds. Let $h:M\rightarrow N$ a homeomorphism.
Does there exist $r:M\rightarrow N$ that is still a homeomorphism and additionaly smooth? Can it be chosen in way that retains a certain similarity to the homeomorphism? Analogously, does there exist a smooth homeomorphism $\ell:N\rightarrow M$ that is similar to $h^{-1}$?
I do not demand that the maps are $r$ and $\ell$ are equal, which would not be possible always. Neither I demand them to be diffeomorphisms.
Thanks in advance