Is it possible to take a limit of map projections (from a sphere to a plane) with ever-smaller distortion factors to get some kind of dendritic limit projection that has zero distortion everywhere? My characterization of an $\epsilon$-distortion map (defined in reverse, i.e. plane to sphere) is:
Let $f:\Bbb R^2\to S^2$. Then $f$ is an $\epsilon$-distortion map projection iff for almost every $x\in S^2$ (except a nullset), there is a $y\in f^{-1}[x]$ such that for any $u,v\in\Bbb R^2$, $$\lim_{t\to0}\frac{\langle f(y+tu)-f(y),f(y+tv)-f(y)\rangle}{t^2}\in[\langle u,v\rangle(1-\epsilon),\langle u,v\rangle(1+\epsilon)].$$
Note in particular that $f$ is not required to be injective; rather every point of $S^2$ has to have some preimage point that is $\epsilon$-distortion. Now I can restate my question as: do $0$-distortion maps exist?