<![CDATA[Rotation numbers and the Jankins-Neumann ziggurat]]>

<![CDATA[Rotation numbers and the Jankins-Neumann ziggurat]]>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)

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