# 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 ${r=0}$.

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:

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.

## 4 thoughts on “Winding map and local injectivity”

1. Tomás says:

Do you know any place where I can find this paper? Or maybe, another reference which contains the same subjetc?

1. L says:

The state of the art is “The local homeomorphism property of spatial quasiregular mappings with distortion close to one” by Kaj Rajala. It’s published in GAFA, and in preprint form is available at http://www.math.jyu.fi/research/pspdf/294.pdf

2. Is the point that the derivative is not required to be continuous? I don’t see why the local homeomorphism conclusion doesn’t follow from the inverse function theorem if the smallest singular value doesn’t vanish.

1. L says:

I should not have omitted the technical part (now inserted as a parenthetical into the post). Pointwise differentiability is not assumed; instead, the map is assumed to be in the Sobolev class W^{1,n} so that the Jacobian determinant works as expected.

If we assume pointwise differentiability with nonvanishing Jacobian, local invertibility holds: we don’t even need continuity of the derivative.
http://terrytao.wordpress.com/2011/09/12/the-inverse-function-theorem-for-everywhere-differentiable-maps/

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