I’m in Melbourne right now, where I recently attended the Hyamfest and the preceding workshop. There were many excellent talks at both the workshop and the conference (more on that in another post), but one thing that I found very interesting is that both Michel Boileau and Cameron Gordon gave talks on the relationships between taut foliations, left-orderable groups, and L-spaces. I haven’t thought seriously about taut foliations in almost ten years, but the subject has been revitalized by its relationship to the theory of Heegaard Floer homology. The relationship tends to be one-way: the existence of a taut foliation on a manifold M implies that the Heegard Floer homology of M is nontrivial. It would be very interesting if Heegaard Floer homology could be used to decide whether a given manifold M admits a taut foliation or not, but for the moment this seems to be out of reach.

Anyway, both Michel and Cameron made use of the (by now 20 year old) classification of taut foliations on Seifert fibered 3-manifolds. The last step of this classification concerns the case when the base orbifold is a sphere; the precise answer was formulated in terms of a conjecture by Jankins and Neumann, proved by Naimi, about rotation numbers. I am ashamed to say that I never actually read Naimi’s argument, although it is not long. The point of this post is to give a new, short, combinatorial proof of the conjecture which I think is “conceptual” enough to digest easily.

The conjecture concerns rotation numbers of circle homeomorphisms. Given an orientation-preserving homeomorphism of the circle \varphi, Poincaré defined the so-called rotation number of \varphi as follows. Lift \varphi to a homeomorphism \tilde{\varphi} of the line, then define \text{rot}(\tilde{\varphi})=\lim_{n\to\infty} \tilde{\varphi}^n(0)/n. Then define \text{rot}(\varphi) to be the reduction of \text{rot}(\tilde{\varphi}) mod \mathbb{Z}.

In fact, the conjecture is about the real-valued rotation numbers of the lifts, and can be stated in the form of a question. Given homeomorphisms a,b of the circle, and lifts \tilde{a},\tilde{b} to homeomorphisms of the line with (real-valued) rotation numbers r,s, what is the maximum (real-valued) rotation number of the product \tilde{a}\tilde{b}? We denote this maximum as R(r,s). For elementary reasons it satisfies R(r+n,s+m)=R(r,s)+n+m for any integers n,m so it suffices to restrict attention to 0\le r,s < 1. It is also elementary to show that R(\cdot,\cdot) is monotone nondecreasing (though not continuous) in both r and s; from the form of the answer it suffices to determine R(r,s) for r,s rational.

In this language, what Jankins and Neumann conjectured, and Naimi proved, is the following:

Theorem: R(r,s) = \max (p_1+p_2+1)/q where the maximum is taken over all rational p_1/q \le r and p_2/q \le s.

We show how to turn this into a combinatorial problem, which can then be solved directly. Given homeomorphisms a,b of the circle, with rotation numbers p_1/q_1 and p_2/q_2 respectively, we can choose periodic orbits x_i for a and y_j for b, so that a(x_i)=x_{i+p_1} and b(y_j) = y_{j+p_2}, indices taken mod q_1 and q_2 respectively. Denote the union of the x_i,y_j by \Sigma.

Now, in place of homeomorphisms a and b consider (discontinuous) maps \alpha and \beta defined by \alpha(\theta) = x_{i+p_1} for \theta \in (x_{i-1},x_i], and similarly \beta(\theta) = y_{j+p_2} for \theta \in (y_{j-1},y_j]. The point is that we can adjust the dynamics of a and b on the complement of the x_i and the y_j respectively without changing their rotation number. Replacing \tilde{a} and \tilde{b} with new \tilde{a}',\tilde{b}' satisfying \tilde{a}'(\theta) \ge \tilde{a}(\theta) and \tilde{b}'(\theta)\ge \tilde{b} for all \theta gives \text{rot}(\tilde{a}'\tilde{b}') \ge \text{rot}(\tilde{a}\tilde{b}). If successive elements of \Sigma are at least \epsilon apart, then providing a'(\theta) \in (x_{i+p_1}-\epsilon/2,x_{i+p_1}] for \theta \in (x_{i-1}+\epsilon/2,x_i] (and similarly for b') the powers of \tilde{\alpha}\tilde{\beta} and \tilde{a}'\tilde{b}' have orbits that stay a bounded distance apart.

So in order to find R(p_1/q_1,p_2/q_2) it suffices to study the rotation numbers of \tilde{\alpha}\tilde{\beta} as above. Evidently, these rotation numbers depend (in a simple way, which we will now describe) only on the circular order of the points x_i,y_j. We encode the circular order of the x_i,y_j by a cyclic word W of X‘s and Y‘s, one X for each x_i, and one Y for each y_j. We define a dynamical system, whose states are the letters of W. The transformation \alpha acts by moving to the right p_1+1 X‘s (including the X we start on, if we start on an X) and the transformation \beta acts by moving to the right p_2+1 Y‘s (including the Y we start on, if we start on a Y). Any W with q_1 X‘s and q_2 Y‘s is said to be admissible for q_1,q_2. For each admissible W the transformation \alpha\beta acting on W has an obvious rotation number, and R(p_1/q_1,p_2/q_2) is the maximum of this rotation number over all admissible W. We illustrate this with an example:

Example: To compute R(1/2,2/3) the admissible 2,3 words are (up to cyclic permutation) XXYYY, XYXYY, and XYYXY. Starting on the last (cyclic) letter, and successively applying \alpha,\beta gives in the first case a rotation number of 1, in the second case a rotation number of 3/2, and in the third case a rotation number of 3/2, so R(1/2,2/3)=3/2.

With this setup established, we now prove the theorem:

Proof: We prove the desired inequality for rational r=p_1/q_1 and s=p_2/q_2. Suppose W is an admissible q_1,q_2 word, for which \alpha\beta has rotation number n/m, and suppose this is maximal over all W, so that R(p_1/q_1,p_2/q_2)=n/m. We can decompose W (up to cyclic permutation) into m subwords U_1U_2\cdots U_m so that if U_i^+ denotes the last letter of U_i, then \alpha\beta(U_i^+) = U_{i+n}^+, indices taken mod m. We can similarly decompose a cyclic permutation of W into subwords V_1V_2\cdots V_m so that \beta\alpha(V_i^+) = V_{i+n}^+, indices taken mod m. We can choose indices so that \alpha(V_i^+) = U_i^+ and \beta(U_i^+) = V_{i+n}^+. Let T_k be the subdivision of W generated by the U and V subdivisions. By the definition of the transformations \alpha and \beta, each U_i^+ is a letter X, and each V_i^+ is a letter Y, so the endpoints are distinct, and the T subdivision has exactly 2m elements. We may permute the letters within each T_k without changing the dynamics, providing we keep the last letter fixed. So we can assume that each T_k is either of the form X^kY^l (if T_k^+ = V_i^+ for some i) or of the form Y^lX^k (if T_k^+ = U_i^+ for some i).

Now suppose some V_i is entirely contained in some U_j. Hence V_i coincides with some T_k and therefore V_i = X^kY^l. We claim we can move the X^k string to the left, past the rightmost string of Y‘s in T_{k-1} (note that T_{k-1}^+ = V_{i-1}^+). Since \beta ignores X‘s, we will still have \beta(U_i^+) = V_{i+n}^+ after this transformation. Moreover, since each interval (V_i^+,U_i^+] contains the same or fewer X‘s after this move, we have \alpha(V_i^+)\ge U_i^+ after this transformation; i.e. for the new word we obtain, the rotation number of \alpha\beta is no smaller than it was for W. So without loss of generality, if V_i is entirely contained in some U_j then we can assume that V_i consists entirely of Y‘s; similarly, any U_j contained entirely in V_k can be assumed to consist entirely of X‘s. But this means that W contains at most m consecutive strings of X‘s and Y‘s, and therefore exactly m (since each U_i^+ is an X and each V_i^+ is a Y), so each V_i is of the form Y^{y_i}X^{x_i}Y^{z_i}. This implies that the U_i^+ and V_j^+ alternate, so that there is a fixed l so that V_{i+l}^+ < U_i^+ < V_{i+l+1}^+ for each i. Now, \alpha(V_i^+) = U_i^+ so p_1 \ge x_{i+1} + x_{i+2} + \cdots + x_{i+l}. Since this is true for every i, and since \sum_i x_i = q_1, we get an inequality p_1/q_1 \ge l/m. Similarly, we have an inequality p_2/q_2 \ge (n-l-1)/m. But R(l/m,(n-l-1)/m) \ge n/m, as one can see by considering the dynamics of \alpha and \beta on the word (XY)^m. qed

This combinatorial language turns out to be quite flexible, and one can push the techniques substantially further; Alden Walker and I are busy writing this up at the moment. One of the nice aspects of this story is that it gives rise to attractive pictures; the graph of R(r,s) for 0\le r,s < 1 is the “ziggurat” appearing in the following figure. The vertical faces of the ziggurat correspond to places where R is not continuous as a function of r,s.

The Jankins-Neumann ziggurat (i.e. the graph of R(\cdot,\cdot) in the unit square)