Ziggurats and rotation numbers

We establish the existence of new rigidity and rationality phenomena in the theory of nonabelian group actions on the circle, and introduce tools to translate questions about the existence of actions with prescribed dynamics into finite combinatorics. A special case of our theory gives a very short new proof of Naimi's theorem (i.e. the conjecture of Jankins-Neumann) which was the last step in the classification of taut foliations of Seifert fibered spaces.


Introduction
This paper introduces new techniques and uncovers a range of new phenomena in the topological theory of nonabelian group actions on the circle. We develop computational tools which let us reduce subtle dynamical questions to finite combinatorics, and then connect these questions back to (hyperbolic) geometry and low-dimensional topology.
1.1. A character variety for Homeo + (S 1 ). Linear actions of finitely generated groups are parameterized by character varieties. Because a character variety is an algebraic variety, the characters (which capture the abstract dynamics of a representation) are polynomial functions, and this makes answering questions about the existence or nonexistence of actions with certain properties computationally tractable.
It is natural to want to generalize this theory to nonlinear actions of finitely generated groups on manifolds. In general there is probably no chance of such a generalization; however, for the case of group actions on circles, we believe some elements of the theory can be developed.
Associated to a linear representation ρ : Γ → G is a trace χ : Γ → R. Given Γ and G, a trace is determined by its values on finitely many g ∈ Γ, and for each g ∈ Γ the value χ(g) is an invariant of the conjugacy class of ρ(g). Associated to a nonlinear representation ρ : Γ → Homeo + (S 1 ) one has the rotation number rot : Γ → R/Z, and for each g ∈ Γ the value rot(g) is a (complete) invariant of the semi-conjugacy class of ρ(g) (i.e. the equivalence relation generated by dynamical semi-conjugacy).
Given a group G and a collection of elements g i and h in G, we would like to know what possible dynamics can be achieved by h under representations of G into Homeo + (S 1 ) for which the g i have prescribed dynamics, either in the form of a constraint on the rotation numbers of the g i , or by imposing the condition that the g i are conjugate to rotations through prescribed angles. A good theory should make it possible to compute these values. In fact, it is one of the main goals of this paper to develop tools to allow one to describe the shadows (i.e the projections under rot) of such "algebraic" subsets of Hom(G, Homeo + (S 1 )), at least for G free, or a free product of cyclic groups.
One significant point of difference between characters and rotation numbers is that it is not true that knowing the rotation numbers of finitely many elements suffices to determine the rest. The following example depends on knowledge of some elementary facts about hyperbolic structures on surfaces; see e.g. [14] for a reference.
Example 1.1. Consider a family of representations of F 2 into PSL(2, R) (which acts on RP 1 ) associated to a family of (incomplete) hyperbolic structures on a oncepunctured torus. We identify F 2 with the fundamental group of this torus, so that the generators a and b correspond to the meridian and longitude, and the commutator [a, b] corresponds to the puncture.
At the complete finite area structure, every element is hyperbolic or parabolic, and therefore every element has rotation number 0 (mod Z). As we deform the image of [a, b] from a parabolic to an elliptic element, rot([a, b]) becomes nonzero. However, for every finite collection of elements not conjugate to a power of [a, b], the image stays hyperbolic, and the rotation number stays zero, for sufficiently small perturbations. Moreover, there is a 2 parameter family of deformations of the hyperbolic structure keeping the cone structure near the puncture fixed, and under such a deformation some elements switch from hyperbolic to elliptic or back.
On the other hand, there is a certain sense in which knowledge of all rotation numbers together with certain homological data does determine a representation, again up to the relation of semi-conjugacy. The precise statement is due to Ghys [9] and is further refined by Matsumoto [15], and can be expressed in terms of the "Euler class in integral bounded cohomology"; since a proper discussion of this would take us too far afield, we do not pursue it here.
If it is not true that finitely many rotation numbers determine the rest, it is nevertheless true that the rotation numbers of (say) the generators strongly constrain the rotation numbers of the other elements, and in quite an interesting way. The main problem on which we focus in this paper is therefore the following: Motivating Question. Given a free group F , and an element w of F , and given values of the rotation numbers of the generators, what is the set of possible rotation numbers of w?
We are especially focussed on the special case that w is a positive word in the generators of F ; i.e. it is contained in the semigroup they generate. In this case the theory simplifies significantly, and we are able to obtain strong results.
1.2. Phase locking and greedy rationals. One of the best known and best studied examples of nonlinear phase locking in dynamics is the phenomenon of Arnol'd tongues: the introduction of nonlinear noise into a family of circle homeomorphisms tends to produce periodic orbits; informally, we call this the phenomenon of greedy rationals. In this paper we are concerned with another manifestation of this phenomenon, in the context of nonabelian group dynamics. We find strong constraints on the dynamics of free group actions on the circle that maximize certain dynamical quantities, and in many case show that these these constraints are powerful enough to guarantee periodic orbits and therefore rational rotation numbers.
1.3. New proofs of old theorems. The classification of taut foliations of Seifert fibered 3-manifolds was the culmination of the work of many people, including Thurston, Brittenham, Eisenbud, Hirsch, Jankins, Neumann and others, and was completed by Naimi [17] by proving an outstanding conjecture of Jankins-Neumann [12]. The conjecture of Jankins-Neumann is equivalent to an analysis of the first non-trivial example in our theory, and our methods lead to a new and very short proof of Naimi's result, thereby embedding this classical work into a new and more powerful context. 1.4. Statement of results. For simplicity, the results in our paper are proved for actions of a free group of rank 2 on the circle. However, the generalization to arbitrary rank is straightforward. Statements of the main theorems for arbitrary rank can be found in the appendix; the proofs are routine generalizations of the proofs given in the paper for the rank 2 case, and are left to the reader. § 2 is devoted to setting up notation and proving some elementary results. We introduce the following notation. Let F be a free group of rank 2 generated by elements a, b. Let Homeo + (S 1 ) ∼ denote the universal central extension of the group of orientation-preserving homeomorphisms of the circle. For ϕ ∈ Homeo + (S 1 ) ∼ , let rot ∼ (ϕ) denote the (real-valued) rotation number.
Given w ∈ F and real numbers r, s, let X(w, r, s) denote the set of values of rot ∼ (w) achieved by representations F → Homeo + (S 1 ) ∼ for which rot ∼ (a) = r and rot ∼ (b) = s, and let R(w, r, s) denote the supremum of the set X(w, r, s). In words, R(w, r, s) is the supremum of rot ∼ (w) under all representations for which rot ∼ (a) = r and rot ∼ (b) = s. It turns out (see Lemma 2.13 and Proposition 2.16) that X(w, r, s) is a compact interval with maximum R(w, r, s) (so that the supremum is really achieved) and minimum −R(w, −r, −s), so we focus almost exclusively on R(w, r, s) throughout the paper.
A word w ∈ F is positive if it does not contain a −1 or b −1 . Positive words are the focus of § 3, and our strongest results are stated and proved for positive words.
The first main result we prove is the Rationality Theorem: Rationality Theorem 3.2. Suppose w is positive. If r and s are rational, so is R(w, r, s). Moreover, if w is not a power of a or b, the denominator of R(w, r, s) is no bigger than the minimum of the denominators of r and of s.
Moreover, we are able to reduce the computation of R(w, r, s) in any case to a finite (albeit complicated) combinatorial problem, which can be solved explicitly, e.g. by computer.
Complementing the rationality theorem is the Stability Theorem: Stability Theorem 3.7. Suppose w is positive. Then R is locally constant from the right at rational points; i.e. for every pair of rational numbers r and s, there is an ǫ(r, s) > 0 so that R(w, ·, ·) is constant on [r, r + ǫ) × [s, s + ǫ). Conversely, if R(w, r, s) = p/q (where p/q is reduced) and the biggest power of consecutive a's in w (resp. b's) is a m (resp. b n ), then R(w, r+1/mq, s) ≥ p/q+1/q 2 (resp. R(w, r, s + 1/nq) ≥ p/q + 1/q 2 ).
We think of this theorem as a nonabelian group theoretic cousin of the phenomenon of Arnol'd tongues, with the "tongues" corresponding to the regions in the r-s plane where R(w, ·, ·) is locally constant. The Stability Theorem shows that the sizes of these tongues decrease with q, but experiments suggest a sharper relationship ǫ ∼ q −1 . This seems very exciting to us, and deserves further exploration.
For the special case of w = ab, the combinatorics becomes simple enough to allow a complete solution. This leads to an elementary proof of the following theorem, first conjectured by Jankins-Neumann and proved by Naimi: w = ab Theorem 3.9. For 0 ≤ r, s < 1 there is an equality R(ab, r, s) = sup p1/q≤r, p2/q≤s (p 1 + p 2 + 1)/q As remarked earlier, this theorem is the last step in the classification of taut foliations of Seifert fibered spaces.
If we fix w and a rational number p/q = rot ∼ (a), the function R(w, p/q, ·) is nondecreasing and takes on a discrete set of values (however, it is typically discontinuous). The next theorem says that this function is continuous from the right, and the points of discontinuity are rational, and can be determined by a finite procedure.
Stairstep Theorem 3.11. Let w be positive, and suppose we are given rational numbers p/q and c/d so that c/d is a value of R(w, p/q, ·) (so necessarily d ≤ q). Then inf{t : R(w, p/q, t) = c/d} is rational, and there is an algorithm to compute it. Moreover, if u/v is this infimal value, R(w, p/q, u/v) = c/d.
Although we are not able to give a precise description of the level sets of R, we are able to give a description of the frontier of the set where R(w, ·, ·) ≥ p/q for a given rational number p/q. This is summarized in the following Isobar Theorem: Isobar Theorem 3.32. Let w be positive. For any rational p/q the set of r, s ⊂ [0, 1] × [0, 1] such that R(w, r, s) ≥ p/q is a finite sided rational polyhedron, whose boundary consists of finitely many horizontal or vertical segments.
One subtle point is that the set where R(w, r, s) = p/q is not in general a finite sided rational polyhedron, and can in fact be extremely complicated. § 4 moves on to the case of arbitrary w. The Rationality Theorem generalizes to words w which are semipositive; i.e. that contain to a −1 or no b −1 (Theorem 4.2).
We conjecture (Conjecture 4.5) that for arbitrary w and rational r, s, R(w, r, s) is rational. Given w, a compactness argument (using circular orders rather than actions) shows that there is a representation with rot ∼ (a) = r and rot ∼ (b) = s realizing rot ∼ (w) = R(w, r, s). Counterfactually, we suppose that rot ∼ (w) is irrational, and then try to modify the dynamics to increase it. This gives rise to a dynamical problem of independent interest called the interval game. Roughly speaking, given a finite collection of homeomorphisms ϕ 1 , ϕ 2 , · · · , ϕ m and an irrational number θ, we would like to find an interval I and an integer n for which R i θ (I + ) is not contained in ϕ j (I) for any 0 ≤ i ≤ n and any j, but R n θ (I + ) ∈ I. Here I + denotes the rightmost point of the interval I, and R θ is the rotation R θ : z → z + θ.
We show (Theorem 4.14 and Theorem 4.15) that the interval game can be solved for generic C 1 diffeomorphisms ϕ i , and for all cases of a single homeomorphism ϕ except when ϕ is itself a rotation R φ . When ϕ is a rotation, the set of pairs (θ, φ) for which the interval game can be won turns out to be an interesting open, dense subset of the unit square, which is described in Theorem 4.16. It follows that for w containing a single a −1 , there are strong constraints on the possible irrational values of R(w, r, s), in terms of w and s, namely that they are contained in the complement of an explicit dense G δ .
We are also concerned throughout this paper with representations satisfying the stronger constraint for which a and b are conjugate to rotations R r and R s respectively. We denote by R(w, r−, s−) the supremum of rot ∼ (w) over representations of this kind. It turns out that for w positive, R(w, r−, s−) is the limit of R(w, r ′ , s ′ ) as r ′ → r and s ′ → s from below. Because of this, we can calculate R(w, r−, s−) to any degree of accuracy, and in many (most) cases, show that it is rational. By a trick, for rational r, s and any word w there is some positive word w ′ (w, r, s) and an explicit integer N (w, r, s) for which R(w, r−, s−) = N + R(w ′ , r−, s−), and therefore we can (numerically) compute R(w, r−, s−) for rational r, s and for arbitrary words w. This achieves our goal of making the theory computationally tractable.

Basic definitions and properties
In the sequel we fix F , a free group on two generators a and b. We denote by Homeo + (S 1 ) the group of orientation-preserving homeomorphisms of S 1 , and by Homeo + (S 1 ) ∼ the group of orientation-preserving homeomorphisms of R covering homeomorphisms of S 1 under the covering projection R → S 1 . There is a central extension 0 → Z → Homeo + (S 1 ) ∼ → Homeo + (S 1 ) → 0 and the kernel of Homeo + (S 1 ) ∼ → Homeo + (S 1 ) is the group of integer translations of R.
We denote by R θ ∈ Homeo + (S 1 ) the homeomorphism z → z + θ; we call R θ the rigid rotation through angle θ. By abuse of notation, we also denote by R θ its lift to Homeo + (S 1 ) ∼ . In the first case, θ should be thought of as an element of S 1 ; in the second, as an element of R. Which meaning is intended should be clear from context.

Rotation number.
We assume the reader is familiar with Poincaré's rotation number; for a basic reference, see e.g. Ghys [10] or Herman [11].
Rotation number is a function rot : Homeo + (S 1 ) → R/Z which admits a real valued lift rot ∼ : Homeo + (S 1 ) ∼ → R. The following proposition summarizes some basic properties of these functions. These properties are all well-known, and we will use them without comment throughout the paper. (1) rot and rot ∼ are conjugacy invariants, and are continuous in the C 0 topology.
(2) A homeomorphism ϕ ∈ Homeo + (S 1 ) has rational rotation number with denominator q if and only if it has a periodic orbit with period q. (3) rot and rot ∼ are homomorphisms when restricted to any cyclic (in fact, amenable) group of homeomorphisms. (4) For any ϕ ∈ Homeo + (S 1 ) ∼ with rot ∼ (ϕ) = r, the map t → rot ∼ (R t • ϕ) defines a continuous, nondecreasing, surjective map from [0, 1] → [r, r + 1] (see [11]). Some more substantial, but nevertheless classical results, concern when a homeomorphism is conjugate to some R θ : Theorem 2.2 (Denjoy (see [11])). If ϕ is C 2 and has irrational rotation number, ϕ is (C 0 ) conjugate to a rigid rotation with the same rotation number. Theorem 2.3 (Herman, Yoccoz [11,22]). There is a subset of the set of irrational numbers of full measure, so that if θ is an element of this set, any ϕ which is C ∞ and is conjugate to R θ , is conjugate by a C ∞ diffeomorphism.
For concreteness, we call an irrational θ as in Herman-Yoccoz theorem a Herman number.

Extremal representations.
The fundamental question we are concerned with in this paper is the following: Question 2.4. Given an element w ∈ F and real numbers r, s, what is the supremum of rot ∼ (ρ(w)) over all representations ρ : F → Homeo + (S 1 ) ∼ for which rot ∼ (ρ(a)) = r and rot ∼ (ρ(b)) = s?
The content of Jankins-Neumann's paper [12] is a partial analysis of this question for the important but very special case w = ab. They formulated a complete conjectural answer, and went a long way towards proving it; their conjecture was subsequently established by Naimi [17].
We discuss this special case in § 3.3 and give a new, much shorter, and more direct proof of Jankins-Neumann's conjecture (see Theorem 3.9).
Remark 2.5. One can ask variations on this question for more complicated groups G and elements w ∈ G; some results in this direction are obtained in [2]. The techniques and results obtained in this paper generalize in a straightforward way to the case of a free group F of rank > 2. We introduce the following notation.
Proof. Pick any lift, then multiply the generators by suitable powers of the center.
The function R has some elementary properties, which we record.   Proof. Changing the orientation of the circle multiplies rotation numbers by −1.
Taking inverses multiplies rotation numbers by −1. The composition of these two operations proves the lemma.
There are a priori estimates on R(w, r, s) in terms of the combinatorics of w.
Lemma 2.11. Suppose w is conjugate into the form w = a α1 b β1 · · · a αm b βm for some m = m(w). Then R(w, r, s) ≤ m(w) + rh a (w) + sh b (w).
Proof. The proof is elementary, given some facts from the theory of stable commutator length (denoted scl, see Definition 3.22); we will return to this connection in § 3.8, and the reader can consult this section or [5] for definitions and details.
The chain w − a ha − b h b is homologically trivial, and its stable commutator length is easily shown to satisfy the inequality scl(w − a ha − b h b ) ≤ m(w)/2. On the other hand, the function rot ∼ is a homogeneous quasimorphism with defect at most 1 (this is a restatement of the Milnor-Wood inequality in the language of quasimorphisms), so the inequality follows from generalized Bavard duality ( [5], Thm. 2.79).
Remark 2.12. Evidently Lemma 2.11 is implied by the stronger inequality If r and s are rational, so is the right hand side of this inequality. Lemma 2.13. For any w, r, s the maximum R(w, r, s) is achieved on some representation.
Proof. Rotation number depends (continuously) only on the circular order derived by the action of the group on any orbit. The space of such circular orders is compact, and so is the subset for which a and b have a specific rotation number (mod Z). By Lemma 2.8 and compactness, rot ∼ (w) is maximized on some circular order (which can be realized in many ways by an action).
Lemma 2.14. For fixed w, the function R(w, r, s) is lower semi-continuous in r and s.
Proof. Lower semi-continuity means that the value of a limit is a least as big as the limit of the values. Let r i , s i → r, s and let ρ i be representations for which the maximum is achieved. Some subsequence of the ρ i converges up to semi-conjugacy to a limit ρ, and rot ∼ (ρ(w)) = lim i rot ∼ (ρ i (w)).
Note that h a (w)r + h b (w)s ∈ X(w, r, s) for all w, r, s. The following proposition shows that X(w, ·, ·) can be recovered from R(w, ·, ·). Proposition 2.16. For any w and fixed r, s, the set X(w, r, s) is a compact interval, with maximum R(w, r, s) and minimum −R(w, −r, −s).
Proof. The set of representations for which the rotation numbers of a and b are prescribed is path-connected. This implies that X(w, r, s) is connected, and therefore an interval. The maximum is R(w, r, s) by definition, and the minimum is −R(w, −r, −s) because changing the orientation of the circle negates rotation number.
2.3. Approximation by smooth diffeomorphisms. For later use, we state and prove some elementary facts about approximating homeomorphisms by diffeomorphisms.
Proof. First of all, it is obvious that ϕ can be C 0 approximated by C ∞ diffeomorphisms ϕ + and ϕ − in such a way that |ϕ(p) − ϕ ± (p)| < ǫ for all p, while ϕ + (p) is to the right of ϕ(p) and ϕ − (p) is to the left of ϕ(p). One way to do this is to approximate the graph of ϕ by smooth graphs from above and below. Evidently . This defines a smooth family interpolating between ϕ − and ϕ + , so some element of this family has the same rotation number as ϕ. Moreover, every element of this family satisfies |ϕ t (p)− ϕ(p)| < ǫ for all p. Lemma 2.18. If both r and s are either rational or Herman numbers, every θ in the interior of X(w, r, s) is realized by a smooth representation. If both r, s are Herman numbers, every θ in the interior of X(w, r, s) is realized by a representation in which a and b are smoothly conjugate to R r and R s respectively.
Proof. By Lemma 2.17, there are smooth representations with rot ∼ (a) = r and rot ∼ (b) = s and for which rot ∼ (w) is as close to any θ in X(w, r, s) as we like; hence we can certainly realize a dense subset of θ in X(w, r, s) by smooth representations. On the other hand, the set of diffeomorphisms whose rotation number is equal to a fixed rational or Herman number is path connected.
Indeed, if ϕ and ϕ ′ are both smooth with the same rational rotation number, they both have a finite orbit with the same dynamics. We can easily find a 1parameter family ϕ t interpolating between them with the same finite orbit, and hence the same rotation number.
If ϕ and ϕ ′ are both smooth with the same irrational Herman rotation number r, they are both smoothly conjugate to R r . The two conjugating maps can be joined by a smooth path of conjugating maps, exhibiting a 1-parameter family ϕ t all with rotation number r (in fact, all conjugate to R r ) interpolating between them.
In conclusion, the set of θ ∈ X(w, r, s) realized by smooth representations is dense and connected. The lemma follows.
Remark 2.19. Shigenori Matsumoto communicated the following short proof that for any r and s (not necessarily rational or Herman irrational), every θ in the interior of X(w, r, s) is realized by a smooth representation. This follows by showing that the space of C ∞ diffeomorphisms with a given rotation number r is pathwise connected. For rational r this is obvious; for irrational r, let f be any smooth diffeomorphism with rot(f ) = r, and define Because r is irrational, for each t there is a unique θ(t) so that rot(f t,θ(t) ) = r; see e.g. [11]. Since the subset {(t, θ(t))} is closed in [0, 1] × S 1 (being the preimage of r under the continuous function rot) it follows that t → θ(t) is continuous.

it contains only a's and b's).
In this section we adhere to the convention that w is positive unless we explicitly say otherwise. In this case we are able to obtain complete and precise answers, and a surprisingly rich structure theory.
The first surprise is the following Rationality Theorem, which says that for r, s rational and w positive, R(w, r, s) is rational, and there is an a priori bound on its denominator. In fact, it turns out that the computation of R(w, r, s) in any given case can be reduced to a finite combinatorial question! Theorem 3.2 (Rationality Theorem). Suppose w is positive. If r and s are rational, so is R(w, r, s). Moreover, if w is not a power of a or b, the denominator of R(w, r, s) is no bigger than the minimum of the denominators of r and of s.
Proof. Consider any action of F on S 1 for which rot ∼ (a) = r and rot ∼ (b) = s. We will show how to modify this action without decreasing rot ∼ (w), until w has a periodic orbit with period no bigger than that of a or b. In fact, the new w will have a periodic orbit which can be taken to biject (in a natural way) with a subset of a periodic orbit of either a or b. This will prove the proposition. We call this method of starting with any representation, and modifying the dynamics without decreasing rot ∼ (w) until w has some desired property, the method of perturbation.
Since r and s are rational, both a and b have finite orbits.
. Denote the union of the x i by Σ x , the union of the y j by Σ y , and the union of both by Σ.
We can modify the dynamics of a on the complement of Σ x (and similarly for b) without changing their rotation numbers. Replacing a by some new a ′ with the property that a ′ (p) ≥ a(p) for all p cannot decrease rot ∼ (w), since only positive powers of a appear in w; a similar statement holds for b.
Define maps α + and β + by α Note that although these maps are not homeomorphisms, they are monotone and therefore still have a well-defined rotation number.
Define w + to be the composition obtained by replacing a and b by α + and β + in the word w. Evidently rot ∼ (w + ) ≥ rot ∼ (w). On the other hand, for any ǫ we can and similarly for b ′ and y j . Set ǫ less than half the distance between distinct elements of Σ. Then there is some initial choice of p so that |(w + ) n (p) − w n (p)| < ǫ for any integer n, and therefore rot ∼ (w + ) = rot ∼ (w).
It remains to estimate the denominator of rot ∼ (w + ). Since w is by hypothesis not a power of a, some conjugate of w ends with a, and therefore the corresponding conjugate of w + takes Σ x into itself. The denominator of rot ∼ (w + ) is the least period of an orbit, and is therefore ≤ q 1 . Interchanging a and b gives the desired estimate.
Note that Theorem 3.2 gives an algorithm to compute R(w, r, s) for positive w and rational r and s. For each configuration of Σ in the circle, the rotation number can be read off from the map w + from Σ to itself. This rotation number only depends on the relative order of the points in Σ; there are only finitely many possible configurations, so by examining each of them in turn we can compute the maximum. This is pursued more systematically in § 3.2. Proof. Let ρ be a representation maximizing rot ∼ (w) for fixed r, s. We may increase r by replacing a with the composition R t • a where R t is a rotation through angle t. By Proposition 2.1 bullet (4) we can prescribe rot ∼ (R t • a) ≥ r.
Since w is positive, replacing a with R t • a cannot decrease rot ∼ (w).
Lemma 3.4. Suppose w is positive. If r = p/q is rational, then R(w, r, s) is rational with denominator ≤ q for all s.
Proof. As s increases in some interval, R(w, r, s) is nondecreasing, by Lemma 3.3.
On the other hand, it is rational with denominator ≤ q for all rational s by Theorem 3.2, and therefore by lower semicontinuity (i.e. Lemma 2.14) it is rational with denominator ≤ q for all s.
3.2. XY words and dynamics. We now study the combinatorics of Σ x and Σ y (recall the notation and setup from the proof of Theorem 3.2). A priori it might be possible that R(w, p 1 /q 1 , p 2 /q 2 ) can only be achieved by some configuration where Σ x ∩ Σ y is nonempty. However, the following lemma shows we do not need to consider such configurations.
is achieved for some configuration where Σ x and Σ y are disjoint.
Proof. Suppose we have a configuration in which x i = y j for some i, j (i.e. in which Σ x and Σ y are not disjoint). Perturb x i slightly so that y j−1 < x i < y j . We apply one of α + or β + in turn and see how our new orbit compares to the old one. After applying some power of α + we might end up at x i . But if we then apply β + then because β + (x i ) = β + (y j ) the new orbit immediately catches up to the old. Conversely, if we apply some power of β + and end up at y j and then apply α + , because α + (y j ) = α + (x i+1 ) the new orbit pulls ahead of the old. In either case, we definitely do not lag, and the rotation number is no smaller in the new configuration.
Because of Lemma 3.5, in the sequel we will assume that Σ x and Σ y are disjoint, and therefore |Σ| = q 1 + q 2 . The configuration of Σ x and Σ y in S 1 can be encoded (up to conjugacy) by a cyclic word W in letters X and Y containing q 1 X's and q 2 Y 's; call such a word admissible for the pair q 1 , q 2 . The number of cyclic words admissible for q 1 , q 2 is exponential in min(q 1 , q 2 ), but for fixed q 1 (say), is polynomial in q 2 .
There is a dynamical system, generated by transformations a and b, whose orbit space is the letters of W , as follows. The element a acts by moving to the right until we read off p 1 + 1 X's, counting the X we start on, if we start on an X. The element b acts by moving to the right until we read off p 2 + 1 Y 's, counting the Y we start on, if we start on a Y . A maximal consecutive string a m "hops" over mp 1 + 1 X's, and a maximal consecutive string b n "hops" over np 2 + 1 Y 's; we refer to these prosaically as a-hops and b-hops in the sequel. Since a and b are monotone with respect to the cyclic order, it makes sense to define the (Q-valued) rotation number R(w, p 1 /q 1 , p 2 /q 2 , W ) for any admissible q 1 , q 2 cyclic word W .
Example 3.6. We compute R(ab, 2/3, 1/2). Up to cyclic permutation, there are 2 admissible 3, 2 words, namely XXXY Y and XXY XY . Each application of a skips over 3 X's, and each application of b skips over 2 Y 's. We apply the letters of w from right to left to compute R; therefore we think of our group acting on S 1 on the left.
For the case XXXY Y , there is an orbit of period 1: and the rotation number is 1.
Complementing the Rationality Theorem is the following Stability Theorem: Theorem 3.7 (Stability Theorem). Suppose w is positive. Then R is locally constant from the right at rational points; i.e. for every pair of rational numbers r and s, there is an ǫ(r, s) > 0 so that Conversely, if R(w, r, s) = p/q (where p/q is reduced) and the biggest power of consecutive a's in w (resp. b's) is a m (resp. b n ), then R(w, r+1/mq, s) ≥ p/q+1/q 2 (resp. R(w, r, s + 1/nq) ≥ p/q + 1/q 2 ).
Proof. By Lemma 2.14 and Lemma 3.3, it follows that R(w, ·, s) is continuous from the right. If s is rational, it takes only finitely many values, by Theorem 3.2. Hence it is locally constant from the right. Hence for any r there is rational r ′ > r with R(w, r ′ , s) = R(w, r, s). Similarly, there is a rational s ′ > s with R(w, r ′ , s ′ ) = R(w, r ′ , s). Monotonicity (i.e. Lemma 3.3) proves the existence of an ǫ as in the statement of the theorem.
Conversely, let W be admissible with R(w, r, s, W ) = p/q, where r = u/v. Each time we read a m we hop over mu + 1 X's. In the course of a periodic orbit, the end of this a-hop lands on q distinct X's in W , which we label X 1 , X 2 , · · · , X q . For some i there are at most ⌊v/q⌋ X's in the "interval" (X i , X i+1 ], by the pigeonhole principle. Let W ′ be obtained from W by replacing each X by X mq . Then by comparing orbits, R(w, u/v + 1/mq, s) ≥ R(w, (umq + v)/vmq, s, W ′ ) ≥ p/q + 1/q 2 , as claimed.
It follows that for all positive w, there is an open, dense set in the r-s plane where R(w, r, s) is locally constant and takes values in Q. This is a new manifestation of the familiar phenomenon of phase locking; it would be very interesting to investigate, for a fixed w, how the maximal ǫ as in Theorem 3.7 depends on r and s. For fixed w, a natural guess (in view of the inequality in the second half of Theorem 3.7) is ǫ ∼ q −1 , where p/q is the locally constant value of R(w, ·, ·). There is some experimental evidence for this, but it seems hard to prove rigorously.
Example 3.8. The inequality in Theorem 3.7 is sharp for every q for w = a m b n , as follows from Theorem 3.9, to be proved in § 3.3.
3.3. The case w = ab. In this section we analyze a particular important special case in detail, namely the case w = ab. We derive a concrete formula for R(ab, p 1 /q 1 , p 2 /q 2 ), and thereby give a new (and much simpler) proof of the main conjecture of Jankins-Neumann [12], first proved by Naimi [17].
Proof. By monotonicity, it suffices to prove the conjecture for rational r, s. So let 0 ≤ p 1 /q 1 < 1 and 0 ≤ p 2 /q 2 < 1 be arbitrary. Let W be a cyclic XY word admissible for q 1 , q 2 with R(ab, p 1 /q 1 , p 2 /q 2 ) = R(ab, p 1 /q 1 , p 2 /q 2 , W ) = n/m. We assume n/m is reduced, so that ab has an orbit of period m on W . We decompose where we denote the rightmost letter of U i by U + i . Each U + i is therefore a Y and each T + i is an X, so the set of endpoints of these words are disjoint. Let V 1 V 2 · · · V 2m be the common refinement.
We now show that we can adjust the letters in W without affecting the dynamics. First of all, any reordering of the letters in each V i which leaves the last letter intact will not affect the dynamics, so without loss of generality we assume that for some j (possibly with y = 0). Next, suppose some T i is entirely contained in some U j , so that T i = V k for some k, and T i = Y y X x . Notice V k−1 is also entirely contained in U j , and V + k−1 = T + i−1 . We claim that moving the (possibly empty) string Y y left to the start of V k−1 will not decrease the rotation number of W . First, we still have a(U + l ) = T + n+l for all l (since a ignores Y 's). Second, we have b(T + l ) ≥ U + l for all l, since the number of Y 's between T + l and U + l is either the same or is decreased by this transformation. This proves the claim.
It follows that whenever we have a string of consecutive T i 's entirely contained in some U j , we can move the Y 's out of the T i 's and to the left side of U j . After finitely many transformations of this kind, we can assume that every U j is of the form Y zj X xj Y yj (where a priori possibly x j and/or z j are 0). Now, we claim that in fact every x j > 0. For, otherwise, there is some U j which consists entirely of Y 's. But then T + n+j−1 = a(U + j−1 ) = a(U + j ) = T + n+j which is absurd. The claim follows. In particular, we can conclude that there is some l so that U + i+l < T + n+i < U + i+l+1 for all i. But from this the theorem follows easily. Since a(U + i ) = T + n+i , from the definition of the a transformation we get an inequality p 1 + 1 ≥ x i + x i+1 + · · ·+ x i+l−1 + 1 and therefore p 1 ≥ l−1 j=0 x i+j . Since this is true for every i, and since m j=1 x j = q 1 we conclude p 1 /q 1 ≥ l/m. Similarly, p 2 /q 2 ≥ (n − l − 1)/m. But R(ab, l/m, (n − l − 1)/m, (XY ) m ) = n/m and the theorem is proved.
The proof of Theorem 3.9 is expressed in terms of the purely combinatorial question of maximizing R(ab, p 1 /q 1 , p 2 /q 2 , W ) over admissible q 1 , q 2 words W . It turns out that for the case of ab there is a simple formula relating R to r which solves the combinatorial question of minimizing R(ab, p 1 /q 1 , p 2 /q 2 , W ) over admissible q 1 , q 2 words W . This is the following duality formula: Proposition 3.10 (Duality formula). Suppose p 1 /q 1 and p 2 /q 2 are reduced fractions. There is a formula r(ab, p 1 /q 1 , p 2 /q 2 ) + R(ab, (q 1 − p 1 − 1)/q 1 , (q 2 − p 2 − 1)/q 2 ) = 2 Proof. Without loss of generality we can assume 0 ≤ p i /q i < 1. We have R(ab, (p 1 − q 1 )/q 1 , (p 2 − q 2 )/q 2 ) = R(ab, p 1 /q 1 , p 2 /q 2 ) − 2. On the other hand, R(ab, (p 1 − q 1 )/q 1 , (p 2 − q 2 )/q 2 ) may be calculated as the maximum, over all W admissible for q 1 , q 2 of n/m, where we alternate between moving left q 1 − p 1 X's and q 2 − p 2 Y 's. Changing the orientation of the circle, this evidently computes −r(ab, (q 1 − p 1 − 1)/q 1 , (q 2 − p 2 − 1)/q 2 ).  3.4. Stairsteps. We have described an algorithm to compute R. However, this algorithm as stated is very inefficient because of the very large number of admissible q 1 , q 2 words for large q i . Given a positive word w and a fixed p 1 /q 1 , the function t → R(w, p 1 /q 1 , t) only takes on finitely many values in the interval t ∈ [0, 1) (for, it is a priori bounded, and rational with denominator ≤ q 1 ). Moreover, we already know that this function is nondecreasing as a function of t, and therefore it is completely specified if we know the finitely many values that are achieved, and the infimal t i at which each value is achieved.
The following theorem says that these finitely many values are rational, that each value is achieved at some minimal t i , and gives an algorithm to compute them. Theorem 3.11 (Stairstep Theorem). Let w be positive, and suppose we are given rational numbers p/q and c/d so that c/d is a value of R(w, p/q, ·) (so necessarily d ≤ q). Then inf{t : R(w, p/q, t) = c/d} is rational, and there is an algorithm to compute it. Moreover, if u/v is this infimal value, R(w, p/q, u/v) = c/d.

Proof.
Given w, p/q and c/d we compute the infimal t such that R(w, p/q, t) ≥ c/d; this will give us the same answer. It suffices to compute the infimum over rational t = u/v; we give an algorithm to do this whose output is evidently rational. Without knowing u and v in advance, we let W be an admissible q, v word for which R(w, p/q, u/v) ≥ c/d is achieved. We write W as XY t1 XY t2 · · · XY tq where the t i are (for the moment) real variables subject to linear constraints t i ≥ 0 and t i = v.
We assume c/d is reduced. Suppose equality is achieved, so that w has a periodic orbit of period d. After replacing w by a cyclic permutation we assume it begins with a string of b's and ends with a string of a's, so that the periodic orbit begins in the terminal string of Y 's. We measure the number of X's we jump over or land on at each step. With each maximal a m string we jump over precisely mp + 1 X's, landing on the last one. With each maximal b m string, if we start at the ith X, we jump over l X's where l is the smallest number such that t i + t i+1 + · · · + t i+l ≥ mu + 1. We can rewrite this condition by saying that l is the biggest number such that t i + t i+1 + · · · + t i+l−1 ≤ mu; the advantage of this reformulation is that this latter inequality is homogeneous. Note that we must allow the possibility l = 0 (if t i > mu), in which case this inequality is vacuous. If we only have R(w, p/q, u/v) ≥ c/d then this inequality still holds, but we do not assume l is the biggest number with this property.
Write w d = b β k a α k · · · b β1 a α1 (if w is primitive, the indices are periodic with period k/d). Then we will apply k strings of a's and b's. Let l i for 1 ≤ i ≤ k be the value of l as above when we apply the string b βi . Then we obtain an equality Then our inequalities are precisely of the form t si + t si+1 + · · · + t si+li−1 ≤ β i u, indices taken mod q.
Since our system of inequalities is homogeneous, linear and defined over Z, we can find a solution in integers iff we can find a solution over the reals. We rescale so that v = 1. Our algorithm then has the following form: First, enumerate all nonnegative integral solutions to k i=1 l i = cq − k i=1 (α i p + 1) (there are only finitely many such solutions, equal to the number of ordered partitions of cq− k i=1 (α i p+1) into k non-negative integers). For each such solution, define s i by the formula s i = i j=1 (α j p + 1) + i−1 j=1 l j . Then let u be the smallest real number (necessarily rational) such that the system of equations { q i=1 t i = 1 and for each 1 ≤ i ≤ k, t i ≥ 0 and t si + t si+1 + · · · + t si+li−1 ≤ β i u} has a solution (indices taken mod q). The smallest u over all such collections of l i is the desired quantity. Evidently it is rational, and achieves the smallest possible value of R(w, p/q, ·) which is ≥ c/d. If c/d is achieved, R(w, p/q, u) = c/d.

An interesting special case of the stairstep theorem is when d = q.
Proposition 3.12. Suppose there is some t for which R(w, p/q, t) = c/q, where p, c are coprime to q. Let w = a α1 b β1 · · · a αm b βm . Then R(w, p/q, t) = c/q for t on some interval [u/nq, u/nq + ǫ) where u/n is the least rational for which Proof. Let W = XY t1 XY t2 · · · XY tq as in the proof of Theorem 3.11. Since w has a periodic orbit of period exactly q, any b-string starting on adjacent X's must land in adjacent Y * strings. This dramatically cuts down on the number of partitions we need to consider; for instance, the l i as in Theorem 3.11 are periodic with period k/d. Because of this, the constraints for the linear programming problem are invariant under cyclic permutation of the variables t i , and by convexity, setting all t i equal gives an extremal solution. The result follows Proposition 3.12 gives rise to the following inequality: Proposition 3.13. For any positive word w of the form w = a α1 b β1 · · · a αm b βm , if R(w, p/q, t) = c/q there is an inequality Proof. We always have R(w, p/q, t) ≥ h a (w)p/q + h b (w)t coming from the representation with a = R p/q and b = R t , so we just need to prove the inequality for t = u/nq as in Proposition 3.12.
We have If q is big compared to m, Proposition 3.13 says that R(w, p/q, t) is very close to h a (w)p/q +h b (w)t, which is achieved for the linear representation a = R p/q , b = R t . Combined with the inequality in Theorem 3.7, we obtain strong constraints on the values of R(w, p/q, ·) achieved, especially if w contains a substring of the form a m with m large.
It is natural to wonder whether the inequality in Proposition 3.13 can be generalized to cases when the denominator of R(w, r, s) is strictly smaller than the denominators of r and s; we return to this question in § 3.7.

3.5.
Speeding up the computation. The algorithm described in the proof of Theorem 3.11 has one big bottleneck, namely the need to enumerate the partitions of cq − k i=1 (α i p + 1) into k non-negative integers l i . This number of partitions is exponential in k, which is itself linear in w and d ≤ q. So the runtime of the algorithm above is a priori exponential in w and q.
Apart from the inequalities q i=1 t i = 1 and t i ≥ 0, we are left with the "sinequalities" which are all of the form t si + t si+1 + · · · + t si+li−1 ≤ β i u. For each i with 1 ≤ i ≤ k let J i denote the "interval" [s i , s i + l i − 1] in the "circle" Z/qZ. A partition L := l 1 , l 2 , · · · , l k of cq− k i=1 (α i p+1) thereby determines a combinatorial configuration of intervals in a circle (weighted by integers β i ), and the minimal u(L) depends only on the combinatorics of this (weighted) configuration.
In fact, we consider vectors of non-negative reals r i so that the weighted sum of indicator functions χ(r) := r i χ Ji is ≥ 1 everywhere in the circle. The s-inequalities imply u ≥ 1/( r i β i ), and linear programming duality gives the equality u(L) = sup riχJ i ≥1 1/( r i β i ). Hence the problem becomes to minimize r i β i subject to r i χ Ji ≥ 1; informally, to cover S 1 as "efficiently as possible" with the intervals J i . If we call the minimum of r i β i the efficiency of a covering, then since u = min L u(L), we seek the partition giving rise to the covering of least efficiency.
A partial partition is a vector K := l 1 , l 2 , · · · , l j with j < k and j i=1 l i < cq− k i=1 (α i p+1); such a partial partition can be extended to a (complete) partition L as above in potentially many ways; we write K < L if L extends K to a complete partition. For a partial partition K, define u(K) = sup 1/( r i β i ) over all coverings of S 1 by intervals J i with 1 ≤ i ≤ j; equivalently, we let u(K) be the smallest u for which there is a feasible solution to the linear programming problem determined by the s-inequalities coming from K. Each additional s-inequality can only reduce the space of feasible solutions, and therefore u(K) ≤ u(L) for every extension K < L. But u = min L u(L). Consequently, if there is a complete partition L ′ with u(K) > u(L ′ ), then u(L) > u(L ′ ) for every K < L, and therefore we can ignore all extensions L of K when computing u.
Example 3.14. Since we assume a priori that u ≤ 1, we must have l i ≤ qβ i for all i.
If we enumerate partitions lexicographically, inequalities as above let us prune the tree of partitions and speed up the computation of u. Figure 2 shows the ziggurat for w = abaab, which is computed using a combination of methods. 3.6. Limits and rotations. We introduce the notation R(w, p 1 /q 1 , p 2 /q 2 −) to mean the limit of R(w, p 1 /q 1 , t) as t → p 2 /q 2 from below. Since R(w, p 1 /q 1 , t) is monotone nondecreasing as a function of t, and takes only rational values with denominators bounded by q 1 , this limit is well-defined and rational, with denominator bounded by q 1 , and is achieved on a semi-open interval of values [u/v, p 2 /q 2 ) where u/v can be determined by Theorem 3.11. We similarly introduce the notation R(w, p 1 /q 1 −, p 2 /q 2 ) and R(w, p 1 /q 1 −, p 2 /q 2 −).
Proposition 3.15. Let w be positive, and suppose we are given rational numbers p 1 /q 1 and p 2 /q 2 . Then R(w, p 1 /q 1 −, p 2 /q 2 ) is the supremum of rot ∼ (w) for all representations with a conjugate to the rotation R p1/q1 and rot ∼ (b) = p 2 /q 2 (and similarly with the roles of a and b interchanged), and R(w, p 1 /q 1 −, p 2 /q 2 −) is the supremum of rot ∼ (w) for all representations with a conjugate to R p1/q1 and b conjugate to R p2/q2 . Proof. We first claim that for any a with rot ∼ (a) < p 1 /q 1 there is some α conjugate to R p1/q1 satisfying α(p) ≥ a(p) for all p. As in the proof of Lemma 2.17 we can produce a smooth family a t interpolating between a − and a + , all C 0 close to a. Some a t with a t (p) ≥ a(p) has irrational rotation number, and is therefore conjugate (by some g) to some R θ with rot ∼ (a) ≤ θ < p 1 /q 1 . If we let α be obtained by conjugating R p1/q1 by g then α satisfies the desired properties.
Conversely, any homeomorphism conjugate to R p1/q1 can be C 0 approximated by homeomorphisms conjugate to R t for any t < p 1 /q 1 .
An important special case is 1−. Because R 1 is central, for any positive w we have R(w, 1−, p/q) = h a (w) + h b (w)p/q and R(w, p/q, 1−) = h a (w)p/q + h b (w). This gives rise to straight lines of slope 1, clearly visible in the graphs of R(ab, ·, ·) and R(abaab, ·, ·) in Figures 1 and 2.
The Stairstep Theorem (i.e. Theorem 3.11) implies that for each w and each p/q there is some smallest u/v with R(w, p/q, u/v) = h a (w)p/q + h b (w) (and similarly with the roles of a and b interchanged). Proposition 3.12 says that u/v = u/nq where u/n is minimized subject to Note that the result does not depend on p; this "explains" why the fringes of the ziggurats appear periodic on every scale near the sloped edges. Solving for integers u and n to minimize u/n is straightforward.
On the other hand, R(abaab, t, 1/2−) = 1 + R(abaab −1 , t, 1/2−) where the notation here simply means representations for which b is conjugate to R 1/2 . Obviously R(abaab −1 , t, 1/2−) ≤ R(ab, t, 2t). On the other hand, by the proof of Theorem 3.9, we know that for t = p/q with q odd, the extremal q, q word for R(ab, p/q, 2p/q) is (XY ) q ; evidently the dynamics of a 2 and b are conjugate in this case by an element which is itself conjugate to R 1/2 , so we obtain an equality R(abaab, p/q, 1/2−) = 1 + R(ab, p/q, 2p/q) for q odd (and in fact by the same reasoning, R(aba n b, p/q, 1/2−) = 1 + R(ab, p/q, cp/q) for q odd and n coprime to q).

Slippery points.
Definition 3.19. Let w be positive. A pair of rational numbers (r, s) is slippery for w if there is a strict inequality R(w, r ′ , s ′ ) < R(w, r−, s−) for all r ′ < r, s ′ < s.
If (r, s) is not slippery, then there are r ′ < r and s ′ < s with R(w, r ′ , s ′ ) = R(w, r−, s−). It follows that this common value is achieved on the entire region [r ′ , r) × [s ′ , s) and is therefore rational. The following conjecture generalizes Proposition 3.13, and is supported by some experimental evidence.
Note that for the "linear" representation in which a = R r and b = R s , we have rot ∼ (w) = h a (w)r + h b (w)s. So this conjecture says that the bigger the denominator of R(w, r, s), the closer the extremal representation (one realizing rot ∼ (w)) must be to the linear representation. The idea behind this conjecture is that the "more nonlinear" an extremal representation is, the more rigid it is, and the smaller the denominator of R(w, r, s). Contrapositively, the bigger the denominator, the less rigid (and hence the more slippery) and the closer to linear. This is only heuristic reasoning (and hand-wavy at that), but it is in sympathy with the Stability Theorem.
This conjecture is very interesting in view of the following consequence: Proposition 3.21. The Slippery Conjecture implies that for any slippery (r, s), we have R(w, r−, s−) = h a (w)r + h b (w)s. In particular, it implies that R(w, r−, s−) is rational for all rational (r, s).
Proof. If (r, s) is slippery, there are r ′ < r, s ′ < s arbitrarily close to r, s for which the denominator of R(w, r ′ , s ′ ) is arbitrarily large. Since R(w, r−, s−) is rational for (r, s) rational and not slippery, the proposition follows.
The Slippery Conjecture has been experimentally checked for several words, up through the range q ≤ 14, which is close to the limit of what can be computed easily. Figures 3 and 4 show the range of experimentally computed values of |p/q − h a (w)r − h b (w)s| for words abaab and abaabbabbbababaab respectively, for q ≤ 14. In each case, the maximal error m/q is precisely achieved for "most" q.
Note that for any w we have R(w, 1/2−, 1/2−) = (h a (w) + h b (w))/2. For, any representation in which both a and b are conjugate to rigid rotations with rotation number 1/2 factors through the infinite dihedral group, which is amenable. On any representation which factors through an amenable group, rotation number becomes  3.8. Immersions and scl. We now describe an unexpected connection to hyperbolic geometry and stable commutator length. For the benefit of the reader, we quickly recall some basic elements of the theory of stable commutator length (for a more substantial introduction see [5]). G is a group, and g ∈ [G, G], the commutator length cl(g) is the least number of commutators in G whose product is g, and the stable commutator length is the limit scl(g) := lim n→∞ cl(g n )/n. If g 1 , g 2 , · · · , g m is a collection of elements with i g i ∈ [G, G] then scl ( g i ) := lim n→∞ cl ( i g n i ) /n. Definition 3.23. If G is a group, a homogeneous quasimorphism is a function φ : G → R satisfying φ(g n ) = nφ(g) for all g ∈ G and n ∈ Z, and for which there is a least non-negative real number D(φ) (called the defect) such that |φ(gh Generalized Bavard Duality ( [5], Thm. 2.79) says that for any G, and for any finite set of elements g i with g i ∈ [G, G], there is an equality where the supremum is taken over all homogeneous quasimorphisms.
Any representation ρ : G → Homeo + (S 1 ) gives rise to a homogeneous quasimorphism on G, namely rotation number. In general, this homogeneous quasimorphism satisfies D(φ) ≤ 1, where generically D(φ) = 1. Therefore stable commutator length can be used to give an upper bound on R. However what is quite surprising is that this upper bound is often sharp, under geometrically meaningful conditions. This is the content of the next theorem.
Theorem 3.24. Let γ w be the unique geodesic representative of the word w in the hyperbolic (q 1 , q 2 , ∞)-orbifold O(q 1 , q 2 ). If γ w virtually bounds an positively immersed subsurface S in O(q 1 , q 2 ) then In general there are inequalities where A(γ w ) is the algebraic area enclosed by γ w , and where scl (denoting stable commutator length) is computed in the group G(q 1 , q 2 ) := a, b | a q1 = b q2 = 1 .
The (unique complete finite area) hyperbolic structure on O(q 1 , q 2 ) gives rise to a representation G → PSL(2, R) which can be thought of as a subgroup of Homeo + (S 1 ). This is covered by a representation H → Homeo + (S 1 ) ∼ . The elements a and b are conjugate to rigid rotations, and for this representation rot ∼ (w) − h a (w)/q 1 − h b (w)/q 2 is equal to A(γ w )/2π (this essentially follows from the Gauss-Bonnet theorem). This gives one inequality.
Generalized Bavard duality (applied to any representation) and the fact that the defect of any rotation quasimorphism is at most 1, gives the upper bound − h b (w)a). If γ w virtually bounds a positively immersed surface S in O, this surface and the rotation quasimorphism are both extremal for w (see [4] for details). In this case, both inequalities are equalities (with area(S) = A(γ w )).
Since H is an amenable extension of G, and H 2 (G; R) = 0, the projection from H to G is an isometry for scl, so scl −h b (w)b). But in G, the elements a and b have finite order, so this last term is equal to scl G (w).
Remark 3.25. The group G(q 1 , q 2 ) is virtually free, and therefore scl is rational and can be computed in polynomial time (for fixed q 1 , q 2 ) by the method of [3] (also see [5], Chapter 4). In fact, an algorithm due to Walker lets one compute scl in polynomial time in word length and in q 1 , q 2 .
Moreover, for fixed q 1 , q 2 , the function w → A(γ w ) is an example of a bicombable function (see [6]), and can be computed in linear time. This makes it practical to actually compute the bounds in Theorem 3.24.
Remark 3.26. The question of which geodesics virtually bound immersed subsurfaces of hyperbolic surfaces is very subtle and difficult. This question is pursued in some detail in [4], and in the special case of the modular orbifold in [7].
3.9. Isobars. In this section we prove a structure theorem for the level sets of R. But first we must prove a couple of lemmas. Proof. A rotation number s achieved by some a as above is said to be feasible for the collection of intervals. It is obvious from the definition that the set of feasible s for a given collection of intervals depends only on the combinatorics.
The I j and their complements partition S 1 into disjoint intervals J i . We assign a variable t i to each J i . Showing s is feasible is equivalent to the feasibility of the linear programming problem defined by the following constraints: (1) each t i is in (0, 1) (2) for each I j the set of t i for which the J i ⊂ I j sums to s (3) the sum of all the t i is 1 The connectedness of the set of feasible s follows from convexity. Rationality follows from the form of the linear programming problem. The proof follows.
Remark 3.30. The set of s feasible for a given finite collection of intervals could easily be empty; for example, if one interval is properly contained in another.
Recalling the definition of X(w, r, s) as in Lemma 2.16, we introduce the notation X(w : t) for the set of r, s for which t ∈ X(w, r, s). Evidently X(w : t) is the intersection of the set of r, s with R(w, r, s) ≥ t with the set of r, s with R(w, −r, −s) ≥ −t.
Lemma 3.31. Let w be positive. X(w : t) is the closure of the set of pairs (r, s) such that there is a representation with a conjugate to a rotation R r and b conjugate to a rotation R s , and with rot ∼ (w) = t.
Proof. Suppose (r, s) ∈ X(w : t). As u varies in (−ǫ, ǫ), the maximum of X(w, r + u, s + u) is continuous from the right, and the minimum is continuous from the left. In order to complete the proof we make two observations.
On the one hand, by Lemma 2.18, for (r, s) irrational Herman numbers, any t in the interior of X(w, r, s) can be achieved by a smooth representation for which a is conjugate to R r and b is conjugate to R s .
On the other hand, taking a = R r and b = R s on the nose, we see that h a (w)r + h b (w)s ∈ X(w, r, s) for all (r, s).
So we have two possibilities: either there is u arbitrarily close to 0 with (r + u, s + u) both Herman irrationals and t in the interior of X(w, r + u, s + u), or else h a (w)r + h b (w)s = t; in either case we are done by one of the two observations above.
We are now ready to prove the main result of this section. Theorem 3.32 (Isobar Theorem). Let w be positive. For any rational p/q the set of r, s ⊂ [0, 1] × [0, 1] such that R(w, r, s) ≥ p/q is a finite sided rational polyhedron, whose boundary consists of finitely many horizontal or vertical segments.
We call the frontier of R(w, r, s) ≥ p/q the p/q isobar. This is part of the frontier of the level set R(w, r, s) = p/q. Note that it is not true in general that the closure of the level set itself is a finite sided polyhedron, even for w = ab. Thus the set of r, s such that R(w, r, s) ≤ p/q can be in general quite complicated. For example, the level set R(ab, ·, ·) = 1 has the line r + s = 1 in its frontier, whereas the level set R(abaab, ·, ·) = 2 has infinitely many segments in its frontier. See Figures 1 and 2. Proof. By the discussion above, it is sufficient to prove that X(w : p/q) is a finite sided rational polygon whose boundary consists of horizontal and vertical segments. By Lemma 3.31 we need to compute the set of r, s for which there are representations with a and b conjugate to rotations through r and s respectively, and rot ∼ (w) = p/q. So consider such a representation.
Let w = a α1 b β1 · · · a αm b βm and let M = α i + β i be the word length of w. By the hypothesis on the rotation number, we can tile [0, p] by qM intervals, each of which is of the form [t, a(t)] or [t, b(t)]. Consider the projection of these qM intervals to the circle; there are a large but finite number of combinatorial types for the image; call such a combinatorial type a partition. For each partition, we compute the set of r feasible for the a-intervals and the set of s feasible for the b-intervals, by Lemma 3.29; the set of (r, s) compatible with both is therefore a rectangle with rational vertices. The union of the interiors of these rectangles over all partitions is dense in X(w : p/q); the theorem follows.    Let w be semipositive (without loss of generality, suppose it contains no a −1 ). If r is rational, so is R(w, r, s). Moreover, the denominator of R(w, r, s) is no bigger than the denominator of r.
Proof. After replacing w by a cyclic conjugate, we can assume it ends with a. Let r = p 1 /q 1 , and let Σ x = ∪ i x i be a periodic orbit for a, so that a(x i ) = x i+p1 . Let α + be defined by α + (θ) = a(x i+1 ) for θ ∈ (x i , x i+1 ], and let w + be obtained by replacing a with α + . Then as in Theorem 3.2 w and w + have the same rotation number, whereas w + takes Σ x to itself, and therefore has a periodic orbit with period ≤ q 1 .

Rationality Conjectures.
In this section we state three conjectures on the rationality of R. These conjectures are related, and we explain how Conjecture 4.5 at least would follow if a certain dynamical problem (the interval game; see § 4.3) always had a positive solution.
Unfortunately, there are instances of the interval game which are unwinnable (i.e. unsolvable); however, it turns out that this dynamical problem generically has a positive solution, and that the exceptions must be quite special. This suggests a program to attack the rationality conjectures.
We adopt the following notational convention, which is consistent with our earlier use for positive w: We would like to prove the following conjectures:   Proof. Actually, the proof is a trick. Let r = p 1 /q 1 and s = p 2 /q 2 . Then a q1 is conjugate to the central element R p1 , and similarly for b q2 . It follows that for any factorization w = w 1 w 2 we have w = w 1 w 2 = w 1 a q1 w 2 z −p1 , and therefore we can write w = w ′ z −N for some sufficiently large integer N , where w ′ is positive, and R(w, p 1 /q 1 −, p 2 /q 2 −) = R(w ′ , p 1 /q 1 −, p 2 /q 2 −) − N .
It follows from Proposition 4.7 that R(w, r−, s−) can be computed, at least for r, s ∈ Q, by the method of § 3.8.
There are at least two classes of (r, s) for which R(w, r−, s−) is known to be rational: (1) at values of (r, s) where R(w, ·, ·) is locally constant, we have R(w, r−, s−) = R(w, r, s) ∈ Q; and (2) at values of (r, s) where the upper bound in Theorem 3.24 is realized, we have R(w, r−, s−) ∈ Q. It is nevertheless true that R(w, r−, s−) can be quite complicated.
Example 4.8. For any r, s (not necessarily rational) we have R(aba −1 b −1 , r−, s−) = 0. To see this, observe that ba −1 b −1 is conjugate to R −r , and therefore there is some point p for which ba −1 b −1 (p) = p − r; but then p is fixed by aba −1 b −1 , which therefore has rotation number 0.
Example 4.9. We now discuss R(aba −1 b −1 , r, s), as an interesting counterpoint to Example 4.8. First of all, we claim that R(aba −1 b −1 , r, s) = 0 whenever r or s is irrational. We argue analogously to the case of Example 4.8: suppose r is irrational, and let µ be an invariant probability measure for a supported on an exceptional minimal set. If there is an interval I with µ(I) = r and µ(b −1 (I)) < r then ≥ µ(I) for every interval I with µ(I) = r then b actually preserves µ, and therefore the action is semiconjugate to a linear action, and rot ∼ (aba −1 b −1 ) = 0.
If r = p/q is rational, then a(·) has a periodic orbit x 1 , x 2 , · · · , x q with indices corresponding to the cyclic order, and a(x i ) = x i+p . If for some i, so that rot ∼ (aba −1 b −1 ) ≤ 1/q. If s = p ′ /q we can build an action which is a q-fold cyclic cover of an action for which both a and b have fixed points; this shows that R(aba −1 b −1 , p/q, p ′ /q) ≥ 1/q and therefore R(aba −1 b −1 , p/q, p ′ /q) = 1/q.
Compare with [20], Remark 3.8. Note that Conjecture 4.4 can only fail for some w, r, s if (r, s) is slippery for w ′ (with notation from the proof of Proposition 4.7). Therefore Conjecture 4.4 is implied by the Slippery Conjecture.
4.3. The Interval Game. We pursue the following strategy to attack Conjecture 4.6, based more or less on the method of perturbation. This leads to a dynamical problem that we call the interval game. The structure of the set of "losing games" is very interesting, even when restricted to a very simple class of games (e.g. consisting entirely of rigid rotations).
The strategy is as follows. Given a word w, and given r, s ∈ Q we suppose that we have a representation for which a and b have the prescribed rotation numbers, and for which rot ∼ (w) is maximal. Suppose that rot ∼ (w) is irrational. We would like to adjust a to a new map a ′ (with the same rotation number as a) which will adjust w to a new w ′ that has a strictly bigger rational rotation number. If we could do this, we would obtain a contradiction, and therefore conclude that R(w, r, s) was rational after all.
We look for a suitable interval I ⊂ S 1 and require a ′ to agree with a outside I, but a ′ > a on I. Providing I can be chosen in the complement of a periodic orbit, a and a ′ will have the same rotation number. The problem is that increasing a on I will decrease a −1 on a(I), and it is not clear if we can find an adjustment for which the net effect on w is to increase its rotation number.
We abstract the situation in terms of a (one-player) game as follows. After conjugating w, we assume that w ends with a, and we let w 1 , w 2 , · · · , w k be the (finitely many) suffixes of w that begin with a −1 (we take suffixes instead of prefixes because our group acts on the left).
Proposition 4.11. Suppose the interval game has a winning interval, for ϕ i = w −1 i and ψ = w, constrained relative to the partition consisting of intervals complementary to a finite orbit for a, and suppose rot ∼ (w) is irrational. Then R(w, r, s) > rot ∼ (w).
Proof. Let I be a winning interval. We adjust a to a ′ on I, and consider the dynamics of a point in I under powers of w ′ . We can build a foliation encoding the dynamics on the mapping torus of w as follows. Let C = ∪C i be a cylinder with a vertical product foliation, decomposed into subcylinders each of which represents the dynamics of one letter of w. As we read w from right to left, the subcylinders from bottom to top represent the dynamics of each successive letter. Then the top of C is glued to the bottom by w.
Adjusting a to a ′ on I can be realized by adjusting the foliation in each subcylinder associated to an a or a −1 in w; see Figure 7. The figure shows the altered dynamics on an a-subcylinder and an a −1 -subcylinder.
Ignoring all a-subcylinders except the first (associated to the terminal a of w) for the moment, and under the hypothesis that I is winning, we see that the future itinerary of any point in I under powers of w ′ is the periodic orbit w(I + ), w 2 (I + ), · · · , w n (I + ), w(I + ); i.e. w ′ has rational rotation number, which is therefore strictly greater than that of w. Taking into account the adjusted dynamics at other a-subcylinders can only further increase the rotation number of w ′ . This completes the proof.
It is therefore important to understand precisely which interval games have a winning interval. First, we recall for the benefit of the reader, some elementary facts about the orbit of a point under successive powers of an irrational rotation. Fix some θ ∈ (0, 1), and a rigid rotation R θ through θ. The case that θ ∈ Q is completely straightforward, so suppose θ is irrational. We express θ as a continued fraction Define θ i recursively by θ 0 = θ, θ 1 = 1 − a 1 θ 0 and θ k+1 = θ k−1 − a k+1 θ k for positive k. Fix r ∈ S 1 , and let r i := R i θ (r) denote the forward orbit. The following lemma is straightforward (see e.g. [16] pp. 26-30): Lemma 4.12. Let r ji for i = 1, 2, · · · be the sequence of successively closest approaches to r; i.e. satisfying |r − r ji | < |r − r k | for k < j i . Then the r ji alternately approach r from the left (if i is odd) and the right (if i is even), and |r − r ji | = θ i .
Moreover, if r a , r b are adjacent elements of {r i | 0 ≤ i ≤ n} with a < b then |r a − r b | = |r − r b−a |; it follows that there are infinitely many odd i for which |r − r ji | < |r a − r b | for all adjacent r a , r b with a, b = 0. Definition 4.13. We say θ is well approximated from the left if there are a sequence of odd i for which |θ i |/|θ i−1 | → 0.
The set of θ ∈ S 1 that are well approximated from the left has full measure; this is elementary, and follows e.g. by the kinds of estimates proved in [11].
The following theorem says that generic interval games (in a suitable sense) have a winning interval.
Theorem 4.14. Consider the interval game associated to a collection ϕ 1 , · · · , ϕ m of C 1 diffeomorphisms, and ψ a rigid rotation through angle θ, where θ is irrational and well approximated from the left.
Suppose there is a point p ∈ S 1 at which the derivatives of the ϕ i are all different from 1. Then there is a winning interval I contained in any subinterval J sufficiently close to p.
Proof. Since the ϕ i are all C 1 , and since their derivatives at p are all different from 1, we can find an interval K contained in J so that the ϕ ′ i are almost constant and bounded away from 1 on K. Intuitively, as we adjust the position of q near p, the ϕ i (q) move almost linearly at speeds bounded away from 1. We can therefore adjust q to a location near p for which none of the ϕ i (q) are too close to some R i θ (q). Let n be very large, and such that R n θ (p) approximates p from the left very well. We fix some very small ǫ, and require that |p − R n θ (p)| ≤ ǫ|R i θ (p) − R j θ (p)| for all 0 < i, j < n. Furthermore, for all 0 < i < n there should be some 0 < j < n not equal to i with |R i θ (p) − R j θ (p)| ≪ |K|. Because the ϕ ′ i are bounded above and below and away from 1, for each i the set of q ∈ K for which |ϕ i (q) − R j θ (q)| ≤ max k∈K |ϕ ′ i (k)| · |q − R n θ (q)| for some j has measure of order ǫ|K|. Choosing ǫ ≪ 1/m we can find some q for which |ϕ i (q) − R j θ (q)| > max k∈K |ϕ ′ i (k)| · |q − R n θ (q)| for all i and all 0 < j < n. Then [R n θ (q), q] is a winning interval. A complete analysis of the interval game seems possible but difficult; however, we are able to completely understand the special case of a single enemy.
Theorem 4.15. Consider the interval game with a single enemy ϕ, and suppose the rotation number of ψ is irrational. Let µ be an invariant probability measure for ψ. If ϕ does not preserve µ, there is a winning interval.
Proof. First for simplicity we suppose that ψ is conjugate to a rigid rotation; equivalently, that µ has full support. The graph Γ of ϕ is a monotone (1, 1) curve in the torus S 1 × S 1 , and by hypothesis, it does not have slope 1 everywhere.
It follows that we can find a slope 1 curve L that locally supports Γ from above, and a point (r, ϕ(r)) which is locally the rightmost point of Γ ∩ L; i.e. Γ is strictly below L in a neighborhood to the right of this point. Dynamically, ϕ is strongly contracting from the right at r; i.e. there is some ǫ so that for |[r, s]| ≤ ǫ, there is an inequality |ϕ([r, s])| < |[r, s]|. In fact, by the strictness of the inequality, there is an ǫ so that for all δ ≤ ǫ there is some s(δ) > r with |ϕ([r, s(δ)])| = |[r, s(δ)]| − δ. Note that the smallest such s(δ) with this property has the additional property that |ϕ([s, s(δ)])| < |[s, s(δ)]| for all r < s < s(δ); i.e. ϕ is strongly contracting from the left at s(δ). Geometrically, if we let L δ denote the line with slope 1 obtained by translating L vertically down δ, then (s(δ), ϕ(s(δ))) is the first time Γ crosses L δ to the right of (r, ϕ(r)); see Figure 8.
We adopt the notation r i := ψ i (r). Since rot(ψ) is irrational, by Lemma 4.12 we can find an arbitrarily big m so that r m is a closest approach to r from the left; in particular, |r − r m | < |r a − r b | for all 0 < a, b < m. Let u = |[r m , r]|.
Let r a , r b , r c , r d be successive orbits such that r c ≤ ϕ(r) < r d . Let t 1 = |[r a , r b ]| and t 2 = |[r b , r c ]|, and v = |[r c , ϕ(r)]|. Note that t 1 > u by the definition of u. We can assume (by taking m sufficiently big) that t 2 + v ≪ ǫ, and therefore there is a point s(t 2 + v) to the right of r with properties as above.
It remains to consider the general case where µ does not have full support. Actually, the argument in this case is essentially the same as that above. In place of Γ we can consider the curve Γ ′ := {( r 0 dµ, ϕ(r) 0 dµ)}. This "graph" might have horizontal and vertical segments, but otherwise we can use the same argument with curves L, L δ applied to Γ ′ in place of Γ. Interestingly enough, the exceptional case that ϕ and ψ are both rigid rotations (after a semiconjugacy) turns out to be nontrivial: Theorem 4.16. Consider the interval game in which both the single enemy ϕ and ψ are rigid rotations through u, t respectively. Then there is a winning interval if and only if (t, u) is contained in an explicit open subset U of the unit square described below (see Figure 9).
Proof. Winning (t, u) are classified by which iterate n of ψ certifies the win, and the smallest integer m such that nt < m. Let U (m) be the subset of U with a given value of m. Then one sees directly that U (1) is the union over all n ≥ 2 and 1 ≤ i ≤ n − 1 of the interior of the triangles with vertices ( 1 n , i n ), ( 1 n−1 , i−1 n−1 ), and ( 1 n−1 , i n−1 ). On the other hand, by rescaling the interval [0, m] by a factor of 1 m it is clear that if ( t m , u+i m ) ∈ U for all 0 ≤ i < m then (t, u) ∈ U , and any element of U (m) is of this kind. This gives a recursive description of U .
The complement of U is the attractor of the IFS generated by the set of projective linear transformations of the form (x, y) → x + n x + n + 1 , y x + n + 1 , (x, y) → x + n x + n + 1 , x + y + n x + n + 1 for all non-negative integers n.
Theorems 4.15 and 4.16 together present an essentially complete picture of the interval game with a single enemy.
Notice for every rational u the set of t with (t, u) in U is an open, dense subset of [0, 1]. As a corollary we get strong constraints on R(w, r, s) for w containing at most one a −1 : Corollary 4.17. Suppose w is a word of the form va −1 b n a for some v containing no a −1 , and suppose r, s ∈ Q. If R(w, r, s) is irrational then (R(w, r, s), ns) is not in U .
In particular, there is a dense G δ subset of [0, 1] that R(w, r, s) avoids. Remark 4.18. For each rational u the set of t with (t, u) not in U is the attractor (i.e. the limit set) of an explicit finitely generated subsemigroup of SL(2, Z) acting projectively on the interval. For example, if u = 1/2, the set of "bad" t is the limit set of the semigroup generated by the matrices If T is a finitely generated semigroup of contractions of the interval whose images are disjoint, and some of the maps have a neutral (i.e. parabolic) fixed point, Urbański ([21]; see also [18]) showed, generalizing work of Bowen [1], that the Hausdorff dimension of the limit set is the least zero of the pressure function P , defined by the formula P (s) = lim where T n is the subset of T consisting of words of length n, and · is the supremum norm. Actually computing this dimension in practice seems hard.

Acknowledgments
We would like to thank Kathryn Mann, Shigenori Matsumoto and the anonymous referee for some useful comments. Danny Calegari was supported by NSF grant DMS 1005246.

Appendix A. Higher rank
For convenience, we state the analogues of our main theorems to higher rank. The proofs of these theorems are routine generalizations of the proofs in the body of the paper, and are omitted.
Throughout this section, let F be a free group of rank n with free generating set a 1 , a 2 , · · · , a n . For real numbers r 1 , r 2 , · · · , r n and w ∈ F , let R(w, r 1 , r 2 , · · · , r n ) denote the maximum of rot ∼ (w) under representations F → Homeo + (S 1 ) ∼ for which rot ∼ (a i ) = r i .
Theorem A.1 (High rank Rationality Theorem). Suppose w is positive. If the r i are all rational, so is R(w, r 1 , · · · , r n ). Moreover, if w has at least one a i , the denominator of R(w, r 1 , · · · , r n ) is no bigger than that of r i .
Theorem A.2 (High rank Stability Theorem). Suppose w is positive. Then R is locally constant from the right at rational points; i.e. for every collection of rational numbers r i , there is an ǫ > 0 so that R(w, · · · ) is constant on [r 1 , r 1 + ǫ) × · · · × [r n , r n + ǫ).
Conversely, if R(w, r 1 , · · · , r n ) = p/q (where p/q is reduced) and the biggest power of consecutive a i s in w is a mi i then there is an inequality R(w, r 1 , · · · , r i + 1/m i q, · · · , r n ) ≥ p/q + 1/q 2 Theorem A.3 (High rank Stairstep Theorem). Let w be positive, and suppose we are given rational numbers p j /q j for j = i and c/d so that R(w, p 1 /q 1 , · · · , t i , · · · , p n /q n ) = c/d for some real t i (so necessarily d ≤ q j for each j). Then the infimum t i with this property is rational, and there is an algorithm to compute it. Moreover, if p i /q i is this infimal value, R(w, p 1 /q 1 , · · · , p n /q n ) = c/d. Theorem A.4 (High rank Isobar Theorem). Let w be positive. For any rational p/q the set of (r 1 , · · · , r n ) ∈ [0, 1] n such that R(w, r 1 , · · · , r n ) ≥ p/q is a finite sided rational polyhedron, whose boundary consists of finitely many polyhedra on which at least one r i is constant.
The results in § 4 also generalize in a straightforward way to higher rank, but we omit the statements.