Winding map and local injectivity

The winding map {W} is a humble example that is conjectured to be extremal in a long-standing open problem. Its planar version is defined in polar coordinates {(r,\theta)} by

\displaystyle    (r,\theta) \mapsto (r,2\theta)

All this map does it stretch every circle around the origin by the factor of two — tangentially, without changing its radius. As a result, the circle winds around itself twice. The map is not injective in any neighborhood of the origin {r=0}.

3D winding
2D winding

The 3D version of the winding map has the same formula, but in cylindrical coordinates. It winds the space around the {z}-axis, like this:

2D winding
3D winding

In the tangential direction the space is stretched by the factor of {2}; the radial coordinate is unchanged. More precisely: the singular values of the derivative matrix {DW} (which exists everywhere except when {r=0}) are {2,1,1}. Hence, the Jacobian determinant {\det DW} is {2}, which makes sense since the map covers the space by itself, twice.

In general, when the singular values of the matrix {A} are {\sigma_1\ge \dots \ge\sigma_n}, the ratio {\sigma_n^{-n} \det A} is called the inner distortion of {A}. The word “inner” refers to the fact that {\sigma_n} is the radius of the ball inscribed into the image of unit ball under {A}; so, the inner distortion compares this inner radius of the image of unit ball to its volume.

For a map, like {W} above, the inner distortion is the (essential) supremum of the inner distortion of its derivative matrices over its domain. So, the inner distortion of {W} is {2}, in every dimension. Another example: the linear map {(x,y)\mapsto (3x,-2y)} has inner distortion {3/2}.

It is known that there is a constant {K>1} such that if the inner distortion of a map {F} is less than {K} almost everywhere, the map is locally injective: every point has a neighborhood in which {F} is injective. (Technical part: the map must be locally in the Sobolev class {W^{1,n}}.) This was proved by Martio, Rickman, and Väisälä in 1971. They conjectured that {K=2} is optimal: that is, the winding map has the least inner distortion among all maps that are not locally injective.

But at present, there is still no explicit nontrivial lower estimate for {K}, for example we don’t know if inner distortion less than {1.001} implies local injectivity.