The normal bundle to the Devil’s staircase and other questions that keep me up at night

It has recently come to my attention that one can define the normal vector field to certain extremely singular “submanifolds” or “subvarieties” of a manifold or variety. I’m using scare quotes here because I’m pretty sure that nobody in their right mind would actually consider such a thing a manifold or variety. In the case of the standard Devil’s staircase (whose construction I will shortly recall) I believe that this vector field should be explicitly computable, though I haven’t been able to figure out how to do it.

Let us begin with the abstract definition of a Devil’s staircase:

A Devil’s staircase is a curve $\gamma: [0, 1] \to M$ in a surface M such that we can find local coordinates (x, y) for M around some point on the curve, such that in those coordinates we can view $\gamma$ as the parametrization of a continuous nonconstant function F such that $F'(x) = 0$ away from a set of Lebesgue measure zero.

In other words, F looks constant, except in infinitesimally small line segments where F grows too fast to be differentiable (or even absolutely continuous).

The standard Devil’s staircase is constructed from the usual Cantor set in $[0, 1]$. To construct the Cantor set C, we start with a line segment, and split it into equal thirds. We then discard the middle third, leaving us with two equal-length line segments. We iterate this process infinitely many times. Clearly we can identify the points that we’re left with with the paths through a full infinite binary tree, so the Cantor set is uncountable[1].

The Cantor set comes with a natural probability measure, called the Cantor measure. One can define it by flipping a fair coin every time you split the interval into thirds. If you flip to heads, you move to the left segment; if you flip to tails, you move to the right segment. After infinitely many coin flips, you’ve ended up at a point in the Cantor set. Thinking of the Cantor set as a subset of $[0, 1]$, you can define the cdf F of the Cantor measure, called the Cantor function:

Choose a random number $0 \leq P \leq 1$ using the Cantor measure. If $x < y$ are real numbers, then the Cantor function F is defined by declaring that $F(y) - F(x)$ is the probability that $x < P \leq y$. The standard Devil’s staircase is the graph of the Cantor function.

It is easy to see that the standard Devil’s staircase is an abstract Devil’s staircase. First, the length of an interval in the nth stage of the Cantor set construction is $3^{-n}$ and there are $2^n$ such intervals; it follows that the Cantor set has length at most $(2/3)^n$. Since n was arbitrary, the Cantor set has Lebesgue measure zero. Outside the Cantor set, we can explicitly compute $F' = 0$. Since F is a cdf, it is a continuous surjective map $F: [0, 1] \to [0, 1]$.

The Devil’s staircase is extremely useful as a counterexample, as it is about as singular as a curve of bounded variation can be, so heuristically, if we want to know if we can carry out some operation on curves of bounded variation, then it should suffice to check on Devil’s staircases.

Let me now construct the normal bundle to the standard Devil’s staircase[2]. For every smooth vector field X on $[0, 1]^2$, we define $\int_{[0, 1]^2} X ~d\omega = \int_{\{u \leq 0\}} \nabla \cdot X$. Then $X \mapsto \int_{[0, 1]^2} X ~d\omega$ can be shown to be bounded on $L^\infty$, so it extends to every continuous vector field on $[0, 1]^2$ and hence defines a covector-valued Radon measure $\omega$ by the Riesz-Markov representation theorem. On the other hand, the divergence theorem says that if an open set $U$ has a smooth boundary, then $\int_U \nabla \cdot X$ is the integral of the normal part of X to $\partial U$. In other words, integrating against $d\omega$ should represent “integrating the part of the vector field which is normal to the Devil’s staircase”.

We can take the total variation $|\omega|$ of $\omega$, and by the Lebesgue differentiation theorem[3], one can show that the 1-form $\alpha(x) = \lim_{r \to 0} \omega(B(x, r))/|\omega|(B(x, r))$ exists for $|\omega|$-almost every x. But $|\omega|$ is the Hausdorff length measure on the Devil’s staircase, and the Devil’s staircase can be shown to have length 2, yet the parts which are horizontal just have length 1. Therefore $\alpha(x)$ must be defined for some x which is not in the horizontal part of the Devil’s staircase. Sharpening $\alpha$, we obtain the normal vector field to the Devil’s staircase.

To see that the sharp of $\alpha$ is really worthy of being called a normal vector field, we first observe that it has length 1 by definition, and second observe that for every vector field X, $\int_\gamma (X, \alpha) ~ds = \int_\gamma (X, \alpha) ~d|\omega| = \int_\gamma X ~d\omega$ where $ds$ is arc length. So pairing against $\alpha$ and then integrating against arc length is integrating the part of the vector field which is normal to the staircase.

The Lebesgue differentiation theorem is far from constructive. So what is the normal vector field to the Devil’s staircase? There should be some nice way to compute the normal field over some point P in the Cantor set in terms of how “dense” the Cantor set is at that point, say in terms of the $(2/3)$-dimensional Hausdorff measure of small balls around P. That, in turn, should be computable in terms of the infinite binary string which defines P. But I don’t know how to do that. I’d love to talk about this problem with you, if you do have an idea.

[1] In fact the Cantor set is a totally disconnected and perfect, compact metrizable space, which characterizes it up to homeomorphism. We could also characterize it up to homeomorphism as the initial object in the category of compact metrizable spaces modulo automorphism.

[2] Actually, the reason that I started looking into this stuff is that I needed to define a normal bundle to extremely singular closed submanifolds of general manifolds. If one wants a definition that does not require a choice of trivialization of the tangent bundle or Riemannian metric, I think one needs the notion of a “bundle-valued Radon measure”. More on that soon…if my definition works.

[3] One needs to use a more general Lebesgue differentiation theorem to do this. In particular, one needs to use the Besicovitch covering lemma in the proof. This raises an interesting question, since the Besicovitch covering lemma has an, apparently, combinatorial constant, which I will call the Besicovitch number. Is there a nice way to compute the Besicovitch number of a Riemannian manifold? Some cute algorithm maybe?