The winding map is a humble example that is conjectured to be extremal in a long-standing open problem. Its planar version is defined in polar coordinates by
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 .
The 3D version of the winding map has the same formula, but in cylindrical coordinates. It winds the space around the -axis, like this:
In the tangential direction the space is stretched by the factor of ; the radial coordinate is unchanged. More precisely: the singular values of the derivative matrix (which exists everywhere except when ) are . Hence, the Jacobian determinant is , which makes sense since the map covers the space by itself, twice.
In general, when the singular values of the matrix are , the ratio is called the inner distortion of . The word “inner” refers to the fact that is the radius of the ball inscribed into the image of unit ball under ; so, the inner distortion compares this inner radius of the image of unit ball to its volume.
For a map, like above, the inner distortion is the (essential) supremum of the inner distortion of its derivative matrices over its domain. So, the inner distortion of is , in every dimension. Another example: the linear map has inner distortion .
It is known that there is a constant such that if the inner distortion of a map is less than almost everywhere, the map is locally injective: every point has a neighborhood in which is injective. (Technical part: the map must be locally in the Sobolev class .) This was proved by Martio, Rickman, and Väisälä in 1971. They conjectured that 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 , for example we don’t know if inner distortion less than implies local injectivity.