GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S^{1}

LIZZIE BURSLEM AND AMIE WILKINSON

Introduction

This paper describes two novel phenomena in the study of infinite group actions on compact manifolds. We exhibit a finitely generated group Γ and a manifold M such that:

• Γ has exactly countably infinitely many effective real-analytic actions on M, up to conjugacy in Diff ^{ω}(M);

• every effective, real analytic action of Γ on M is C^{r} locally rigid, for some r ≥ 3, and for every such r, there are infinitely many nonconjugate, effective real-analytic actions of Γ on M that are C^{r} locally rigid, but not C^{r−1} locally rigid.

In the cases we know of where an infinite group Γ has exactly countably many smooth effective actions on a manifold M, that countable number is finite, and indeed usually 0. While many actions have been shown to to be C^{r} locally rigid, in the cases where a precise cutoff in rigidity has been established, it occurs between r = 1 and r = 2. For a survey of some of the recent results on smooth group actions, see the paper of Labourie [La].

Our manifold M is the circle S^{1} and our group Γ is the solvable Baumslag-Solitar group:

BS(1, n) =< a, b, | aba^{−1} = b^{n} >,

where n ≥ 2.

As a natural by-product of our techniques, we obtain a classification of all solvable subgroups of Diff ^{ω}(S^{1}). We show that every such subgroup G is either conjugate in Diff ^{ω}(S^{1}) to a subgroup of a ramified affine group Aff^{s}(ℝ), or, for some m ∈ ℤ, the group G^{m} := {g^{m} : g ∈ G} is abelian. The ramified affine groups are defined and their properties discussed in Section 2.2. Each ramified affine group is abstractly isomorphic to a direct product Aff_{+}(ℝ) × H, where Aff_{+}(ℝ) is the group of orientation-preserving affine transformations of ℝ, and H is a subgroup of a finite dihedral group.

1. Statement of results

1.1. Notation and preliminary definitions. In places, we shall use two different analytic coordinatizations of the circle S^{1}. To denote an element of the additive group, ℝ/ℤ, we will use u, and for an element of the real projective line ℝP^{1}, we will use x. These coordinate systems are identified by: u ∈ ℝ/ℤ ↦ x = tan(πu) ∈ ℝP^{1}. When we are not specifying a coordinate system, we will use p or q to denote an element of S^{1}. We fix an orientation on S^{1} and use “<” to denote the counterclockwise cyclic ordering on S^{1}.

If G is a group, then we denote by R^{r}(G) the set of all representations ρ_{0} : G → Diff ^{r}(S^{1}), and we denote by R_{+}^{r}(G) the set of all orientation-preserving representations in R^{r}(G). Two representations ρ_{1}, ρ_{2} ∈ R^{r}(G) are conjugate (in Diff ^{r}(S^{1})) if there exists h ∈ Diff ^{r}(S^{1}) such that, for every γ ∈ G, hρ_{1}(γ)h^{−1} = ρ_{2}(γ).

Date: May 17, 2004.

We use the standard C^{k} topology on representations of a finitely-generated group into Diff ^{r}(S^{1}), r ∈ {1, …, ∞, ω} and k ≤ r. If Γ is a finitely-generated group, then the C^{k}- open sets in R^{r}(Γ) take generators in a fixed generating set for Γ into C^{k}-open sets. A representation ρ_{0} ∈ R^{r}(Γ) is (C^{r}) locally rigid if there exists a C^{1} neighborhood U of ρ_{0} in R^{r}(Γ) such that every ρ ∈ U is conjugate in Diff ^{r}(S^{1}) to ρ_{0}. Finally, we say that Γ is globally rigid in Diff ^{r}(S^{1}) if there exists a countable set of locally rigid representations in R^{r}(Γ) such that every faithful representation in R^{r}(Γ) is conjugate to an element of this set.

To construct the subgroups and representations in this paper, we use a procedure we call real ramified lifting.

Definition: A real analytic surjection π : S^{1} → S^{1} is called a ramified covering map over p ∈ S^{1} if the restriction of π to π^{−1}(S^{1} \ {p}) is a regular analytic covering map onto S^{1} \ {p} of degree d ≥ 1. The degree of of π is defined to be this integer d.

Examples and properties of ramified covering maps and ramified lifts are described in Section 2.

Let π : S^{1} → S^{1} be a ramified covering map over p ∈ S^{1}, and let f : S^{1} → S^{1} be a real analytic diffeomorphism that fixes p. We say that f̂ ∈ Diff ^{ω}(S^{1}) is a π-ramified lift of f if the following diagram commutes:

[figure: S^{1}; S^{1}; S^{1}; S^{1}; f̂; f; π; π]

More generally, let ρ : Γ → Diff ^{ω}(S^{1}) be a representation with global fixed point p. A representation ρ̂ : Γ → Diff ^{ω}(S^{1}) is a π-ramified lift of ρ if ρ̂(γ) is a π-ramified lift ρ(γ), for every γ ∈ Γ. We will show in Proposition 2.3 that a representation can have more than one π-ramified lift.

For G a subgroup of Diff ^{ω}(S^{1}) with a global fixed point p, we define Ĝ^{π}, the π-ramified lift of G to be the collection of all π-ramified lifts of elements of G. By Proposition 2.3 and Propostion 2.8, Ĝ^{π} is a subgroup of Diff ^{ω}(S^{1}), abstractly isomorphic to an H-extension of G_{+}, where G_{+} = Diff _{+}^{ω}(S^{1}) ∩ G, and H is a subgroup of a dihedral group determined by π.

1.2. Rigidity of solvable Baumslag-Solitar groups. In real projective coordinates on ℝP^{1}, the standard representation ρ_{n} of BS(1, n) into Diff _{+}^{ω}(S^{1}) takes the generators a and b to the affine maps

x ↦ nx, and x ↦ x + 1.

This representation has a global fixed point ∞ ∈ ℝP^{1}. Our first result states that BS(1, n) is globally rigid in Diff ^{ω}(S^{1}):

Theorem 1.1. For each n ≥ 2, there are exactly countably infinitely many faithful representations of BS(1, n) into Diff ^{ω}(S^{1}), up to conjugacy in Diff ^{ω}(S^{1}). Each conjugacy class contains a π-ramified lift of ρ_{n}, where π : ℝP^{1} → ℝP^{1} is a rational map that is ramified over ∞. Furthermore, if ρ : BS(1, n) → Diff ^{ω}(S^{1}) is not faithful, then there exists a k ≥ 1 such that ρ(b)^{k} = id.

We give an explicit description of these conjugacy classes in Section 2.1.

The conclusion of Theorem 1.1 does not hold when C^{ω} is replaced by a lower differentiability class such as C^{∞}, even when analytic conjugacy is replaced by topological conjugacy in the statement. Nonetheless, as r increases, there is a sort of “quantum rigidity” phenomenon. Let ρ : BS(1, n) → Diff ^{2}(S^{1}) be a representation, and let f = ρ(a). We make a preliminary observation:

Lemma 1.2. If the rotation number of f is irrational, then g^{k} = id, for some k ≤ n + 1, where g = ρ(b).

(See the beginning of Section 5 for a proof). Hence, if ρ ∈ R^{2}(BS(1, n)) is faithful, then f must have periodic points. For ρ ∈ R^{2}(BS(1, n)) a faithful representation, we define the inner spectral radius σ(ρ) by:

σ(ρ) = sup\{|(f^{k})′(p)|^{\frac{1}{k}} | p ∈ Fix(f^{k}) and |(f^{k})′(p)| ≤ 1\}.

For the standard representation, σ(ρ_{n}) = \frac{1}{n}, and if ρ̂_{n} is a ramified lift of ρ_{n}, then σ(ρ̂_{n}) =

(\frac{1}{n})^{\frac{1}{s}}, for some s ∈ ℕ_{≥1}.

Theorem 1.3. Let ρ : BS(1, n) → Diff ^{r}(S^{1}) be a faithful representation, where r ∈ [2, ∞].

If either r < ∞ and σ(ρ) ≤ (\frac{1}{n})^{\frac{1}{r−1}} , or r = ∞ and σ(ρ) < 1, then ρ is conjugated by an element of Diff ^{r}(S^{1}) into a unique conjugacy class in R^{ω}(BS(1, n)).

If ρ takes values in Diff _{+}^{r}(S^{1}), then ρ is conjugated by an element of Diff _{+}^{r}(S^{1}) into a unique conjugacy class in R_{+}^{ω}(BS(1, n)).

Theorem 1.3 has the following corollary:

Corollary 1.4. Every representation ρ : BS(1, n) → Diff ^{ω}(S^{1}) is C^{∞} locally rigid. Fur-

ther, if σ(ρ) < (\frac{1}{n})^{\frac{1}{r−1}} , then ρ is C^{r} locally rigid.

This corollary implies that the standard representation is C^{r} locally rigid, for all r ≥ 3, and every representation in R^{ω}(BS(1, n)) is locally rigid in some finite differentiability classes. This local rigidity breaks down, however, if the differentiability class is lowered.

Proposition 1.5. For every representation ρ : BS(1, n) → Diff ^{ω}(S^{1}), if σ(ρ) = (\frac{1}{n})^{\frac{1}{r−1}} , for some r ≥ 2 then there exists a family of representations ρ_{t} ∈ R^{r}(BS(1, n)), t ∈ (−1, 1), with the following properties:

(1) ρ_{0} = ρ,

(2) t ↦ ρ_{t} is continuous in the C^{r−1} topology on R^{r}(BS(1, n)),

(3) for every t_{1}, t_{2} ∈ (−1, 1), if ρ_{t_{1}} is conjugate to ρ_{t_{2}} in Diff ^{1}(S^{1}) then t_{1} = t_{2}.

It follows from our characterization of the conjugacy classes in R^{ω}(BS(1, n)) in the next section that, for each value of r ∈ [1, ∞), there are infinitely many nonconjugate

representations ρ ∈ R^{ω}(BS(1, n)) satisfying σ(ρ) = (\frac{1}{n})^{\frac{1}{r}} . Hence, for each r ≥ 3 there are infinitely many distinct (nonconjugate) representations in R^{ω}(BS(1, n)) that are C^{r} locally rigid, but not C^{r−1} locally rigid.

A. Navas has given a complete classification of C^{2} solvable group actions, up to finite index subgroups and topological semiconjugacy. One corollary of his result is that every faithful C^{2} representation ρ of BS(1, n) into Diff ^{2}(S^{1}) is is virtually topologically semiconjugate to the standard representation:

Theorem 1.6. [Nav] Let ρ : BS(1, n) → Diff ^{r}(S^{1}) be a representation, where r ≥ 2. Then either ρ is unfaithful, in which case ρ(b)^{m} = id, for some m, or there exists an integer m ≥ 1, a finite collection of closed, connected sets I_{1}, …, I_{k}, and a surjective continuous map φ : S^{1} → ℝP^{1} with the following properties:

(1) ρ(b)^{m} is the identity on each set I_{k}.

(2) φ sends each set I_{k} to ∞;

(3) the restriction of φ to S^{1} \ ⋃_{i=1}^{k} I_{i} is a C^{r} covering map of ℝ;

(4) For every γ ∈ BS(1, n), the following diagram commutes:

[figure: S^{1}; S^{1}; S^{1}; S^{1}; ρ(γ^{m}); ρ_{n}(γ^{m}); φ; φ]

where ρ_{n} : BS(1, n) → Diff ^{ω}(S^{1}) is the standard representation.

The map φ in Theorem 1.6 is a sort of “broken C^{r} ramified cover.” The regularity of φ at the preimages of the point ∞ can be poor, and the map can be infinite-to-one on the sets I_{1}, …, I_{k}, but a map with these features is nothing more than a deformation of a ramified covering map. Combining Theorem 1.6 with Theorem 1.3 and the proof Proposition 1.5, we obtain:

Corollary 1.7. Let ρ : BS(1, n) → Diff ^{r}(S^{1}), be any representation, with r ≥ 2. Then either:

(1) ρ is not faithful, and there exists an m ≥ 1 such that ρ(b)^{m} = id;

(2) ρ admits C^{r−1} deformations as in Proposition 1.5; or

(3) ρ is C^{r} conjugated into a unique conjugacy class in R^{ω}(BS(1, n)).

Since the statement of Theorem 1.6 does not appear explicitly in Navas’s paper, and we don’t use this result elsewhere in the paper, we sketch the proof at the end of Section 5.

Finally, note that the trivial representation ρ_{0}(a) = ρ_{0}(b) = id is not rigid in any topology; it can be approximated by the representation ρ(b) = id, ρ(a) = f, where f is an any diffeomorphism close to the identity. Another nice consequence of Navas’s theorem is that this is the only way to C^{2} deform the trivial representation.

Corollary 1.8. There is a C^{2} neighborhood U ⊂ R^{2}(BS(1, n)) of the trivial representation such that, for all ρ ∈ U, ρ(b) = id.

Proof: Let ρ be a C^{2} representation. Since ρ(b) is conjugate by ρ(a) to ρ(b)^{n}, it will have rotation number of the form \frac{k}{n−1} if ρ(a) is orientation-preserving, and of the form \frac{k}{n+1} if ρ(a) is orientation-reversing. Therefore, if ρ is sufficiently C^{0}-close to ρ_{0} and if ρ(b)^{m} = id, for some m ≥ 1, then m = 1. So we may assume that there exists a map φ as in Theorem 1.6 and that m = 1. On a component of S^{1} \ ⋃ I_{i}, φ is a diffeomorphism conjugating the action of ρ to the restriction of the standard representation ρ_{n} to ℝ (in general φ fails to extend to a diffeomorphism at either endpoint of ℝ). But in the standard action, the element ρ_{n}(a) has a fixed point in ℝ of derivative n. If ρ is sufficiently C^{1} close to ρ_{0}, this can’t happen. □

We remark that, in contrast to the results in this paper, there are uncountably many topologically distinct faithful representations of BS(1, n) into Diff ^{ω}(ℝ) (see [FF], Proposition 5.1). The proof of our results uses the existence of a global fixed point on S^{1} for a finite index subgroup of BS(1, n); such a point need not exist when BS(1, n) acts on ℝ. Farb and Franks [FF] studied actions of Baumslag-Solitar groups on the line and circle. Among their results, they prove that if m > 1, the (nonsolvable) Baumslag-Solitar group:

BS(m, n) =< a, b | ab^{m}a^{−1} = b^{n} >,

has no faithful C^{2} actions on S^{1} if m does not divide n. They ask whether the actions of B(1, n) on the circle can be classified. This question inspired the present paper.

1.3. Classification of solvable subgroups of Diff ^{ω}(S^{1}). Several works address the properties of solvable subgroups of Diff ^{r}(S^{1}); we mention a few here. Building on work of Kopell [Ko], Plante and Thurston [PT] showed that any nilpotent subgroup of Diff ^{2}(S^{1}) is in fact abelian. Ghys [Gh1] proved that any solvable subgroup of Diff ^{ω}(S^{1}) is metabelian, i.e., two-step solvable. In the same work, he remarks that there are solvable subgroups of Diff ^{∞}(S^{1}) that are not metabelian. The subgroups he constructs contain infinitely flat elements – nontrivial diffeomorphisms g ∈ Diff ^{∞}(S^{1}) with the property that for some p ∈ S^{1}, g(p) = p, g′(p) = 1, and g^{(k)}(p) = 0 for all k ≥ 2.

Navas [Nav] constructed further examples of solvable subgroups of Diff ^{∞}(S^{1}) with arbitrary degree of solvability, again using infinitely flat elements. As mentioned above, Navas’s work also contains a topological classification of solvable subgroups of Diff ^{2}(S^{1}). As part of a study of ergodicity of actions of discrete groups on S^{1}, Rebelo and Silva [RS] also study the solvable subgroups of Diff ^{ω}(S^{1}).

Our main result in this part of the paper, Theorem 1.9, implies that any solvable subgroup of Diff ^{∞}(S^{1}) that does not contain infinitely flat elements is either virtually abelian or conjugate to a subgroup of a ramified lift of the affine group:

Aff(ℝ) = {x ↦ cx + d : ℝP^{1} → ℝP^{1} | c, d ∈ ℝ, c ≠ 0}.

Theorem 1.9. Let G < Diff ^{r}(S^{1}) be a solvable group, where r ∈ {∞, ω}. Then either: (1) for some m ∈ ℤ, the group G^{m} := {g^{m} : g ∈ G} is abelian,

(2) G contains infinitely flat elements (which can’t happen if r = ω), or

(3) G is conjugate in Diff ^{r}(S^{1}) to a subgroup of a π-ramified lift of Aff(ℝ), where

π : ℝP^{1} → ℝP^{1} is a ramified cover over ∞. Further, if G < Diff _{+}^{r}(S^{1}) and (3) holds, then the conjugacy can be taken in Diff _{+}^{r}(S^{1}).

In Section 2, we characterize the ramified lifts of Aff(ℝ). To summarize the results there, we have:

Theorem 1.10. There exists a collection

RAFF := \{Âff^{s}(ℝ) < Diff ^{ω}(S^{1}) | s ∈ \overline{S}\},

where \overline{S} is a countably infinite index set, with the following properties:

(1) if s_{1}, s_{2} ∈ \overline{S} and Âff^{s_{1}}(ℝ) is conjugate to Âff^{s_{2}}(ℝ) in Diff ^{1}(S^{1}), then s_{1} = s_{2};

(2) for each s ∈ \overline{S}, there exists a subgroup H of a dihedral group such that Âff^{s}(ℝ) ≃ Aff_{+}(ℝ) × H,

(3) for each finite dihedral or cyclic group H, there exist infinitely many s ∈ \overline{S} so that Âff^{s}(ℝ) ≃ Aff_{+}(ℝ) × H,

(4) each element of RAFF is the π-ramified lift of Aff(ℝ), for some rational ramified cover π : ℝP^{1} → ℝP^{1} over ∞, and every π-ramified lift of Aff(ℝ) is conjugate in Diff ^{ω}(S^{1}) to an element of RAFF.

There also exists a collection

RAFF_{+} := \{Âff_{+}^{s}(ℝ) < Diff _{+}^{ω}(S^{1}) | s ∈ \overline{S}_{+}\},

with the same properties, except that in (1) and (4), the conjugacy is orientation-preserving, and in (2) and (3), H is cyclic.

Hence we have found all solvable groups that act effectively on the circle as real-analytic diffeomorphisms.

2. Introduction to ramified lifts

Let G be a group and let ρ : G → Diff ^{ω}(S^{1}) be a representation with a global fixed point p. Restricting each element of this representation to a suitably small neighborhood of p, we obtain a representation ρ̃ : G → G^{ω}, where G^{ω} is the group of analytic germs of diffeomorphisms. It is known [CM, Nak, EISV] that if if G is solvable, then for some k ≥ 1, ρ̃ is conjugate in G^{ω} to a representation taking values in the ramified affine group Aff ^{k}(ℝ):

Aff ^{k}(ℝ) = \{\frac{x}{(ax^{k} + b)^{\frac{1}{k}}} | a, b ∈ ℝ, b > 0\}

(see [Gh1] for a proof in the context of circle diffeomorphisms). The name “ramified affine group” is explained by the fact that the elements of Aff ^{k}(ℝ) are lifts of the elements of the affine group under the branched (or ramified) cover z ↦ z^{k}. These lifts are well-defined as holomophic germs, but do not extend to diffeomorphisms of ℂP^{1}.

The key observation of this paper is that the elements of Aff(ℝ) do admit global ramified lifts as diffeomorphisms of ℝP^{1}. The reason is that, in contrast to a ramified cover π : ℂP^{1} → ℂP^{1}, which must be ramified over 2 points, a ramified cover π : ℝP^{1} → ℝP^{1} is ramified over one point, which can be chosen to coincide with the global fixed point of Aff(ℝ).

Examples of real ramified covers. The map π_{1} : ℝ/ℤ → ℝ/ℤ given by π_{1}(u) = sin^{2}(πu) is a ramified covering map over 0, with critical points of order 2 at π_{1}^{−1}(0) = \{0, \frac{1}{2}\}.

The rational map π_{2} : ℝP^{1} → ℝP^{1} given by:

π_{2}(x) = \frac{(x + 1)^{2}(x − 1)^{2}}{x(x^{2} + 1)}

is also a ramified covering map over 0, with critical points of order 2 at ±1. It is clear that ±1 are critical points of π_{2}, and one verifies directly that the other critical points of π_{2} in ℂP^{1} occur off of ℝP^{1}, at ±i\sqrt{3 ± \sqrt{8}}.

We will define an equivalence relation on ramified covering maps in which π_{1} and π_{2} are equivalent, and show that, under this notion of equivalence, all possible ramified covering maps occur as rational maps.

If π : S^{1} → S^{1} is a ramified covering map over p then for each q ∈ π^{−1}(p), there exists an integer s(q) ≥ 1 such that the leading (nonconstant) term in the Taylor expansion of π at q is of order s(q). A regular covering map is a ramified covering map; in this case, d is the topological degree of the map, and s(q) = 1, for each q ∈ π^{−1}(p). As the examples show, a ramified covering map need not be a regular covering map (even topologically), as it is possible to have s(q) > 1.

Let π be a ramified covering map over p, and let q_{1}, …, q_{d} be the elements of π^{−1}(p), ordered so that p ≤ q_{1} < q_{2} < ⋯ < q_{d} < p. For each i ∈ {1, …, d} we define o_{i} ∈ {±1} by:

o_{i} = \begin{cases}1 & if π|_{(q_{i},q_{i+1})} is orientation-preserving, \\ −1 & if π|_{(q_{i},q_{i+1})} is orientation-reversing. \end{cases}

We call the vector s(π) = (s(q_{1}), s(q_{2}), …, s(q_{d}), o_{1}, … o_{d}) ∈ ℕ^{d} × {±1}^{d} the signature of π. Geometrically, we think of a signature as a regular d-gon in ℝ^{2} with vertices labelled by s_{1}, …, s_{d} and edges labelled by o_{1}, …, o_{d}. Every signature vector s = (s_{1}, …, s_{d}, o_{1}, …, o_{d}) has the following two properties:

(1) The number of vertices with an even label is even: #{1 ≤ i ≤ d | s_{i} ∈ 2ℕ} ∈ 2ℕ.

(2) If a vertex has an odd label, then both edges connected to that vertex have the same label, and if a vertex has an even label, then the edges have opposite labels: (−1)^{s_{i}+1} = o_{i−1} o_{i}, where addition is mod d.

We will call any vector s ∈ ℕ^{d} × {±1}^{d} with these properties a signature vector. Note that a signature vector of length 2d is determined by its first d + 1 entries. Let S_{d} be the set of all signature vectors with length 2d, and let S be the set of all signature vectors.

Proposition 2.1. Given any s ∈ S and p ∈ S^{1}, there is a ramified covering map π : S^{1} → S^{1} over p with signature s.

Proof: Let s = (s_{1}, …, s_{d}, o_{1}, …, o_{d}) be a signature, and let p ∈ ℝ/ℤ. Choose points u_{1} < ⋯ < u_{d} evenly spaced in ℝ/ℤ, and let F : ℝ/ℤ → ℝ/Z be the piecewise affine map that sends the u_{i} to p, and which sends each component of ℝ/ℤ \ {u_{1}, …, u_{d}} onto ℝ/Z \ {p}, with orientation determined by o_{i}.

Put a new analytic structure on ℝ/ℤ as follows. In the intervals I_{j} = (u_{j}, u_{j+1}) use the standard analytic charts, but in each interval J_{j} = (u_{j} − ϵ, u_{j} + ϵ) compose the standard chart (that identifies u_{j} with 0) with the homeomorphism σ_{j} : ℝ → ℝ defined by:

σ_{j}(x) = \begin{cases}x^{1/s_{j}} & if x > 0 \\ −|x|^{1/s_{j}} & if x ≤ 0. \end{cases} .

Since the overlaps are analytic, this defines a real anaytic atlas on ℝ/ℤ.

Note that the map F : (ℝ/ℤ, new structure) → (ℝ/ℤ, standard structure) is analytic: in charts around u_{j} and p = F (u_{j}), the map F takes the form x ↦ x^{s_{j}}. Since there is a unique real analytic structure on the circle, there is an analytic homeomorphism of the circle h : (ℝ/ℤ, standard structure) → (ℝ/ℤ, new structure). Let π = F ∘ h. Then π is a a ramified covering map over 0 with signature s. □

In fact, ramified covers exist in the purely algebraic category; every signature can be realized by a rational map. We have:

Proposition 2.2. Given any s ∈ S and p ∈ ℝP^{1}, there is a rational map π : ℝP^{1} → ℝP^{1} that is a ramified cover over p with signature s.

Proof: Since the proof of Proposition 2.2 is somewhat lengthy, we omit the details. The construction proceeds as follows. Let s = (s_{1}, …, s_{d}, o_{1}, …, o_{d}) be a signature, and assume that p = 0 ∈ ℝP^{1} and o_{1} = 1. Choose a sequence of real numbers a_{0} < a_{1} < … < a_{2d−2}, let P (x) = (x − a_{0})^{s_{1}}(x − a_{2})^{s_{2}} … (x − a_{2d−2})^{s_{d}} and let Q(x) = (x − a_{1})(x − a_{3}) … (x − a_{2d−3}). The desired rational function π will be a modification of P/Q.

Let h(x) be a polynomial of even degree with no zeros, with critical points of even degree at a_{i}, 0 ≤ i ≤ 2d − 2, and with no other critical points. One first shows that, for N sufficiently large, the rational function:

π_{0} = \frac{P h^{N}}{Q}

has zeroes of order s_{1}, …, s_{d} at a_{0}, a_{2}, …, a_{2d−2}, simple poles at a_{1}, a_{3}, …, a_{2d−3}, a pole of odd order at ∞, and no other zeroes, poles or critical points. Hence π_{0} is a ramified covering map over 0 with signature s, except at ∞, where it may fail to be a diffeomorphism.

Choose such an N, and let 2m + 1 be the order of the pole ∞ for π_{0}. One then shows that for ε sufficiently small, the rational function:

π(x) = \frac{π_{0}(x)}{1 + εx^{2m}}

has the same properties as π_{0}, except that ∞ is now a simple pole; it is the desired ramified cover. □

If π is a ramified covering map, then the cyclic and dihedral groups:

C_{d} = ⟨b : b^{d} = id⟩, and D_{d} = ⟨a, b : b^{d} = id, a^{2} = id, aba^{−1} = b^{−1}⟩,

respectively, act on π^{−1}(p) and on the set E(π) of oriented components of S^{1} \ π^{−1}(p) in a natural way. By an orientation-preserving homeomorphism, we identify the circle with a regular oriented d-gon, sending the elements of π^{−1}(p) to the vertices and the elements of E(π) to the edges. The groups C_{d} ◁ D_{d} act by symmetries of the d-gon, inducing actions on π^{−1}(p) and E(π) that are clearly independent of choice of homeomorphism. For q ∈ π^{−1}(p), e ∈ E(π), and ζ ∈ D_{d}, we write ζ(q) and ζ(e) for their images under this action.

These symmetry groups also act on the signature vectors in S_{d} in the natural way, permuting both vertex labels and edge labels. For ζ ∈ D_{d}, we will write ζ(s) for the image of s ∈ S_{d} under this action. In this notation, the action is generated by:

b(s_{1}, …, s_{d}, o_{1}, …, o_{d}) = (s_{2}, s_{3}, …, s_{d}, s_{1}, o_{2}, o_{3}, …, o_{d}, o_{1}),

and

a(s_{1}, …, s_{d}, o_{1}, …, o_{d}) = (s_{1}, s_{d}, s_{d−1} …, s_{3}, s_{2}, −o_{d}, −o_{d−1}, …, −o_{2}, −o_{1}).

Denote by Stab_{C_{d}}(s) and Stab_{D_{d}}(s) the stabilizer of s in C_{d} and D_{d}, respectively, under this action:

Stab_{H}(s) = {ζ ∈ H | ζ(s) = s},

for H = C_{d} or D_{d}.

Examples: The signature vector of π_{1}(u) = sin^{2}(πu) is s_{1} = (2, 2, 1, −1). The stabilizer of s_{1} in D_{d} is Stab_{D_{d}}(s_{1}) =< a >, and the stabilizer of s_{1} in C_{d} is trivial. The signature vector of π_{2}(x) = ((x − 1)^{2}(x + 1)^{2})/(x(x^{2} + 1)) is s_{2} = (2, 2, −1, 1). Note that s_{2} lies in the C_{d}-orbit of s_{1}, and so Stab_{C_{d}}(s_{2}) and Stab_{D_{d}}(s_{2}) must be conjugate to Stab_{C_{d}}(s_{1}) and Stab_{D_{d}}(s_{1}), respectively, by an element of C_{d}. In this simple case, the stabilizers are equal.

For another example, consider the signature vector (2, 3, 1, 2, 3, 1, −1, −1, −1, 1, 1, 1), which geometrically is represented by the following labelled graph:

[figure: 2; 1; 3; 2; 1; 3; −; −; −; +; +; +]

This labelling has no symmetries, despite the fact that the edge labels have a flip symmetry and the vertex labels have a rotational symmetry. By contrast, the signature (2, 1, 4, 2, 1, 4, −1, −1, 1, −1, −1, 1) has a 180 degree rotational symmetry corresponding to the element b^{3} ∈ C_{6}, and so both stabilizer subgroups are < b^{3} >.

2.1. Characterization of ramified lifts of the standard representation ρ_{n} of BS(1, n). The next proposition gives the key tool for lifting representations under ramified covering maps.

Proposition 2.3. Let G be a group, and let ρ : G → Diff _{+}^{ω}(S^{1}) be a representation with global fixed point p. Let π : S^{1} → S^{1} be a ramified covering map over p with signature s ∈ S_{d}, for some d ≥ 1.

Then for every homomorphism h : G → Stab_{D_{d}}(s), there is a unique representation

ρ̂ = ρ̂(π, h) : G → Diff ^{ω}(S^{1})

such that, for all γ ∈ G, (1) ρ̂ is a π-ramified lift of ρ;

(2) ρ̂(γ)(q) = h(γ)(q), for each q ∈ π^{−1}(p);

(3) ρ̂(γ)(e) = h(γ)(e), for each oriented component e ∈ E(π); Furthermore, if h takes values in Stab_{C_{d}}(s), then ρ̂ takes values in Diff _{+}^{ω}(S^{1}).

Proposition 2.3 is a special case of Proposition 4.4, which is proved in Section 4. Note that the representation ρ in Proposition 2.3 must be orientation preserving, although the lift ρ̂ might not be, depending on where the image of h lies. There is also a criterion for lifting representations into Diff ^{ω}(S^{1}) that are not necessarily orientation-preserving. We discuss this issue in the next subsection.

Lemma 2.4. Suppose that π_{1} and π_{2} are two ramified covering maps over p such that s(π_{2}) lies in the D_{d}-orbit of s(π_{1}); that is, suppose there exists ζ ∈ D_{d} such that s(π_{2}) = ζ(s(π_{1})). Then given any representation ρ : G → Diff _{+}^{ω}(S^{1}) with global fixed point p and any homomorphism h : G → Stab_{D_{d}}(s_{1}), the representations ρ̂(π_{1}, h) and ρ̂(π_{2}, ζhζ^{−1}) are conjugate in Diff ^{ω}(S^{1}), where

(ζhζ^{−1})(γ) := ζh(γ)ζ^{−1}.

Furthermore, if ζ ∈ C_{d} and h takes values in Stab_{C_{d}}(s), then ρ̂(π_{1}, h) and ρ̂(π_{2}, ζhζ^{−1}) are conjugate in Diff _{+}^{ω}(S^{1}).

Lemma 2.4 follows from Lemma 4.5, which is proved in Section 4. We now characterize the countably many conjugacy classes in R^{ω}(BS(1, n)). Note that the elements of S_{d} are totally ordered by the lexicographical order on ℝ^{n}. Hence we can write S_{d} as a disjoint union of C_{d}-orbits:

S_{d} = ⨆_{α∈A_{+}} C_{d}(s_{α}),

where for each α ∈ A_{+}, s_{α} is the smallest element in its C_{d}-orbit. Similarly, there is an index set A ⊃ A_{+} such that:

S_{d} = ⨆_{α∈A} D_{d}(s_{α}).

Let \overline{S}_{d} = \{s_{α} | α ∈ A\}, and let \overline{S}_{d}^{+} = \{s_{α} | α ∈ A_{+}\}. Finally, let \overline{S} = ⋃_{d} \overline{S}_{d} and let \overline{S}^{+} = ⋃_{d} \overline{S}_{d}^{+}.

Definition: Let ρ_{n} : BS(1, n) → Diff _{+}^{ω}(S^{1}) denote the standard projective action, with global fixed point at ∞ ∈ ℝP^{1}. Then we define:

V = \{ρ̂_{n}(π_{s}, h) | s ∈ \overline{S}_{d}, h ∈ Hom (BS(1, n), Stab_{D_{d}}(s))/ ≡, d ∈ ℕ, d ≥ 1\},

and let

V_{+} = \{ρ̂_{n}(π_{s}, h) | s ∈ \overline{S}_{d}^{+}, h ∈ Hom (BS(1, n), Stab_{C_{d}}(s)), d ∈ ℕ, d ≥ 1\},

where, for s ∈ S_{d}, π_{s} : S^{1} → S^{1} is the rational ramified cover over ∞ with signature s given by Proposition 2.2, and ≡ denotes conjugacy in Stab_{D_{d}}(s).

Proposition 2.5. Each element of V and V_{+} represents a distinct conjugacy class of faithful representations.

That is, if ρ̂_{n}(π_{s_{1}}, h_{1}), ρ̂_{n}(π_{s_{2}}, h_{2}) ∈ V (resp. ∈ V_{+}) are conjugate in Diff ^{1}(S^{1}) (resp. in Diff _{+}^{1}(S^{1})), then s_{1} = s_{2} and h_{1} = h_{2}.

Proposition 2.5 is proved at the end of Section 4. Our main result, Theorem 1.1, states that the elements of V and V_{+} are the only faithful representations of BS(1, n), up to conjugacy in Diff ^{ω}(S^{1}) and Diff _{+}^{ω}(S^{1}), respectively.

2.2. Proof of Theorem 1.10. To characterize the ramified lifts of Aff(ℝ), we need to define ramified lifts of orientation-reversing diffeomorphisms. In the end, our description is complicated by the following fact: in contrast to lifts by regular covering maps, ramified lifts of orientation-preserving diffeomorphisms can be orientation-reversing, and vice versa.

To deal with this issue, we introduce another action of the dihedral group D_{d} =< a, b, | a^{2} = 1, b^{d} = 1, aba^{−1} = b^{−1} > on S_{d} that ignores the edge labels completely. To distinguish from the action of D_{d} on S_{d} already defined, we will write ζ^{#} : S_{d} → S_{d} for the action of an element ζ ∈ D_{d}. In this notation, the action is generated by:

b^{#}(s_{1}, …, s_{d}, o_{1}, …, o_{d}) = (s_{2}, s_{3}, …, s_{d}, s_{1}, o_{1}, o_{2}, …, o_{d}),

and

a^{#}(s_{1}, …, s_{d}, o_{1}, …, o_{d}) = (s_{1}, s_{d}, s_{d−1} …, s_{3}, s_{2}, o_{1}, o_{2}, …, o_{d}).

For s ∈ S_{d}, we denote by Stab_{D_{d}}^{#}(s) and Stab_{C_{d}}^{#}(s) the stabilizers of s in D_{d} and C_{d}, respectively under this action.

Lemma 2.6. For each s ∈ S_{d}, there exists a homomorphism

Δ_{s} : Stab_{D_{d}}^{#}(s) → ℤ/2ℤ,

such that Stab_{D_{d}}(s) = ker(Δ_{s}).

Proof: Let s ∈ S_{d} be given. Clearly Stab_{D_{d}}(s) is a subgroup of Stab_{D_{d}}^{#}(s). Let I : S_{d} → S_{d} be the involution:

I(s_{1}, …, s_{d}, o_{1}, …, o_{d}) = (s_{1}, …, s_{d}, −o_{1}, …, −o_{d}).

We show that for every ζ ∈ Stab_{D_{d}}^{#}(s), either ζ(s) = s (so that ζ ∈ Stab_{D_{d}}(s)) or ζ(s) = I(s). This follows from the property (2) in the definition of signature vector, which implies that every element of S_{d} is determined by its first d + 1 entries. Hence we may define Δ_{s}(ζ) to be 0 if ζ(s) = s and 1 otherwise. Since I is an involution, Δ_{s} is a homomorphism. □

Example: Consider the signature s = (2, 1, 2, 1, 2, 1, 2, 1, 1, 1, −1, −1, 1, 1, −1, −1). For this example we have Stab_{C_{8}}(s) =< b^{4} >, Stab_{D_{8}}(s) =< a, b^{4} >, Stab_{C_{8}}^{#}(s) =< b^{2} >, and Stab_{D_{8}}^{#}(s) =< a, b^{2} >. In this example the homomorphism Δ_{s} is surjective, with nontrivial kernel. For s = (2, 1, 4, 1, 2, 1, 4, 1, 1, 1, −1, −1, 1, 1, −1, −1), on the other hand, the image of Δ_{s} is trivial, and Stab_{D_{8}}(s) = Stab_{D_{8}}^{#}(s) =< a, b^{4} >, Stab_{C_{8}}(s) = Stab_{C_{8}}^{#}(s) =< b^{4} >.

For a third example, recall that the stabilizer Stab_{D_{6}}(s) of the signature vector s = (2, 3, 1, 2, 3, 1, −1, −1, −1, 1, 1, 1) is trivial. Because of the rotational symmetry of the vertex labels, however, Stab_{C_{6}}^{#}(s) = Stab_{D_{6}}^{#}(s) =< b^{3} >≃ ℤ/2ℤ. In this example, Δ_{s} is an isomorphism.

Let G < Diff ^{ω}(S^{1}) be a subgroup with global fixed point p ∈ S^{1}: f(p) = p, for all f ∈ G. We now show how to assign, to each s ∈ S, a subgroup Ĝ^{s} consisting of ramified lifts of elements of G. We first write G = G_{+} ⊔ G_{−}, where G_{+} = G ∩ Diff _{+}^{ω}(S^{1}) is the kernel of the homomorphism O : G → ℤ/2ℤ given by:

O(f) = \begin{cases}0 & if f is orientation-preserving, \\ 1 & otherwise. \end{cases}

Suppose that π : S^{1} → S^{1} is a ramified covering map over p. Then, for every f ∈ G_{+}, Proposition 2.3 implies that for every ζ ∈ Stab_{D_{d}}(s(π)), there is a unique lift f̂(π, ζ) ∈ Diff ^{ω}(S^{1}) satisfying:

(1) f̂(π, ζ) is a π-ramified lift of f,

(2) f̂(π, ζ)(q) = ζ(q), for all q ∈ π^{−1}(p), and

(3) f̂(π, ζ)(e) = ζ(e), for all e ∈ E(p) (Further, this lift is orientation-preserving if ζ ∈ Stab_{C_{d}}(s).) Suppose, on the other hand, that f ∈ G_{−}. In Section 4, we prove Lemma 4.2, which implies that if ζ ∈ Stab_{D_{d}}^{#}(s) satisfies:

ζ(s) = I(s), (1)

then there exists a unique lift f̂(π, ζ) ∈ Diff ^{ω}(S^{1}) satisfying (1)-(3) (and, further, f̂(π, ζ) ∈ Diff _{+}^{ω}(S^{1}) if ζ ∈ Stab_{C_{d}}^{#}(s).) We can rephrase condition (1) as:

Δ_{s}(ζ) = 1.

To summarize this discussion, we have proved the following:

Lemma 2.7. If f ∈ G, and ζ ∈ Stab_{D_{d}}^{#}(s), then there exists a lift f̂(π, ζ) satisfying (1)-(3) if and only if:

O(f) = Δ_{s}(ζ).

For s ∈ S, let π_{s} be the ramified covering map over p with signature s given by Proposition 2.1. If G < Diff ^{ω}(S^{1}) has global fixed point p, we define Ĝ^{s} ⊂ Diff ^{ω}(S^{1}) to be the fibered product of G and Stab_{D_{d}}^{#}(s) with respect to O and Δ_{s}:

Ĝ^{s} := {f̂(π_{s}, ζ) | (f, ζ) ∈ G × Stab_{D_{d}}^{#}(s), O(f) = Δ_{s}(ζ)}.

Similarly, we define:

Ĝ_{+}^{s} := {f̂(π_{s}, ζ) | (f, ζ) ∈ G × Stab_{C_{d}}^{#}(s), O(f) = Δ_{s}(ζ)}.

Lemma 2.7 tells us that Ĝ^{s} coincides with Ĝ^{π_{s}}, the set of all π_{s}-ramified lifts of G, and, similarly, that Ĝ_{+}^{s} = Ĝ^{π_{s}} ∩ Diff _{+}(S^{1}). It follows from Lemma 4.3 that Ĝ^{s} and Ĝ_{+}^{s} are subgroups of Diff ^{ω}(S^{1}) and Diff _{+}^{ω}(S^{1}), respectively, with:

f̂_{1}(π_{s}, ζ_{1}) ∘ f̂_{2}(π_{s}, ζ_{2}) = { \atop f_{1} ∘ f_{2}}(π_{s}, ζ_{1}ζ_{2}).

Further, we have:

Proposition 2.8. Assume that G_{−} is nonempty. Then Ĝ^{s} and Ĝ_{+}^{s} are both finite extensions of G_{+}; there exist exact sequences:

1 → G_{+} → Ĝ^{s} → Stab_{D_{d}}^{#}(s) → 1, (2)

and

1 → G_{+} → Ĝ_{+}^{s} → Stab_{C_{d}}^{#}(s) → 1. (3)

Furthermore, if the sequence

1 → G_{+} → G →^{O} O(G) → 1

splits, where O : G → ℤ/2Z is the orientation homomorphism, then the sequences (2) and (3) are split, and so is the sequence

1 → Ĝ_{+}^{s} → Ĝ^{s} → O(Ĝ^{s}) → 1. (4)

Proof: The maps in the first sequence (2) are given by:

ι : G_{+} → Ĝ^{s} f ↦ f̂(π_{s}, id)
σ : Ĝ^{s} → Stab_{D_{d}}^{#}(s)  f̂(π_{s}, ζ) ↦ ζ.

It is easy to see that ι is injective and σ is surjective. Moreover, f̂(π_{s}, ζ) is in the kernel of σ if and only if ζ = id, if and only if f is orientation-preserving, if and only if f̂(π_{s}, id) is in the image of ι. Hence the first sequence is exact. Similarly, the second sequence (3) is exact.

Now suppose that 1 → G_{+} → G → O(G) → 1 is split exact. If O(G) is trivial then G_{+} = G, and there is nothing to prove. If O(G) = ℤ/2ℤ, then G contains an involution g ∈ G_{−} with g^{2} = id, namely the image of 1 under the homomorphism O(G) → G. We use g to define a homomorphism j : Stab_{D_{d}}^{#}(s) → Ĝ^{s} as follows:

j(ζ) = \begin{cases}îd(π_{s}, ζ) & if Δ_{s}(ζ) = 0 \\ ĝ(π_{s}, ζ) & if Δ_{s}(ζ) = 1 \end{cases} .

Hence the sequence (2) is split. The restriction of j to Stab_{C_{d}}^{#}(s) splits the sequence (3).

If O(Ĝ^{s}) is trivial, then the last sequence (4) is trivially split. If O(Ĝ^{s}) = ℤ/2ℤ, then there exists a ζ ∈ Stab_{D_{d}}^{#}(s) such that Δ_{s}(ζ) = 1. We then define k : O(Ĝ^{s}) → G^{s} by k(0) = id, k(1) = ĝ(π_{s}, ζ), which implies that (4) is split. □

Setting G = Aff(ℝ), which has the global fixed point ∞ ∈ ℝP^{1}, we thereby define Âff^{s}(ℝ) and Âff_{+}^{s}(ℝ), for s ∈ S. Let \overline{S} and \overline{S}_{+} be the sets of signatures defined at the end of the previous subsection. The elements of \overline{S} and \overline{S}_{+} are representatives of distinct orbits in S under the dihedral and cyclic actions, respectively. We now define:

RAFF := \{Âff^{s}(ℝ) | s ∈ \overline{S}\}, and RAFF_{+} := \{Âff_{+}^{s}(ℝ) | s ∈ \overline{S}_{+}\}.

Since Aff(ℝ) contains the involution x ↦ −x, the sequence

1 → Aff_{+}(ℝ) → Aff(ℝ) →^{O} ℤ/2ℤ → 1

splits. Proposition 2.8 implies that

Âff^{s}(ℝ) ≃ Aff_{+}(ℝ) × Stab_{D_{d}}^{#}(s),  Âff_{+}^{s}(ℝ) ≃ Aff_{+}(ℝ) × Stab_{C_{d}}^{#}(s),

and either Âff^{s}(ℝ) = Âff_{+}^{s}(ℝ) or Âff^{s}(ℝ) ≃ Âff_{+}^{s}(ℝ) × ℤ/2ℤ, depending on whether Δ_{s} is surjective. This proves (2) of Theorem 1.10.

Corollary 4.7 implies that Âff^{s_{1}}(ℝ) and Âff^{s_{2}}(ℝ) are conjugate subgroups in Diff ^{1}(S^{1}), only if s_{1} ∈ D_{d}s_{2}. Similarly, Âff_{+}^{s_{1}}(ℝ) and Âff_{+}^{s_{2}}(ℝ) are conjugate subgroups in Diff _{+}^{1}(S^{1}), if and only if s_{1} ∈ C_{d}s_{2}. This proves (1) of the theorem.

Finally, every finite dihedral or cyclic group H is the stabilizer of infinitely many s ∈ \overline{S}, and hence there are infinitely many s ∈ \overline{S} so that Âff^{s}(ℝ) ≃ Aff_{+}(ℝ) × H. This proves part (3). Property (4) follows from the definition of RAFF. This completes the proof of Theorem 1.10.

3. The relation fgf^{−1} = g^{λ}: the central technical result

In this section we analyze the relation fgf^{−1} = g^{λ} near a common fixed point of f and g. If f and g are real-analytic, then they can be locally conjugated into one of the ramified affine groups described at the beginning of Section 2 [CM, Nak, EISV]. This gives a local characterization of diffeomorphisms of S^{1} satisfying this relation about each common fixed point, and to obtain a global characterization, it is a matter of gluing together these local ones. This was the way Ghys [Gh1] proved that every solvable subgroup of Diff ^{ω}(S^{1}) is metabelian. To prove Theorems 1.1 and 1.3, we adapt the arguments in [Nak] to the C^{r} setting, where additional hypotheses on f and g are required.

The initial draft of this paper contained a completely different proof of the local characterization of [CM, Nak, EISV] that does not rely on vector fields and works for C^{r} diffeomorphisms as well, under the right assumptions. In the C^{ω} and C^{∞} case, this original proof gives identical results as the vector fields proof, but in the general C^{r} setting, the proof using vector fields gives sharper results. At the end of this section, we outline the alternate proof method. The main idea behind this method is to study the implications of the relation fgf^{−1} = g^{λ} for the Schwarzian derivative of g near a common fixed point at 0.

We now state the main technical result of this section. Let [q, q_{1}) be a half open interval, let r ∈ [2, ∞] ∪ {ω}, and let f, g ∈ Diff ^{r}([q, q_{1})) be diffeomorphisms. Assume that g has no fixed points in (q, q_{1}).

Standing Assumptions. We assume that either A. B. C. or D. holds:

A. r = ω, and there exists an integer λ > 1 such that fgf^{−1} = g^{λ}.

B. r ∈ [2, ∞), and there is an integer λ > 1 such that fgf^{−1} = g^{λ}, and f′(q) ≤ (\frac{1}{λ})^{\frac{1}{r−1}} . C. r = ∞, f′(q) < 1, and for some integer λ > 1, fgf^{−1} = g^{λ}. D. r ∈ {∞, ω}, g is not infinitely flat, and there is a C^{∞} flow g^{t} : [q, q_{1}) → [q, q_{1}) such

that:

(1) g^{1} = g, and

(2) fgf^{−1} = g^{λ} for some positive real number λ ≠ 1.

Assumptions A. B. and C. will arise in the proof of Theorems 1.1 and 1.3, and assumption D. will arise in the proof of Theorem 1.9. The main technical result that we will use in these proofs is the following.

Proposition 3.1. Assume that either A. B. C. or D. holds. Then there is a C^{r} diffeomorphism α : (q, q_{1}) → (−∞, ∞) ⊂ ℝP^{1} such that for all p ∈ (q, α^{−1}(0)):

(1)

α(p) = ϵh(p)^{s},

where h : [q, α^{−1}(0)) → [−∞, 0) is a C^{r} diffeomorphism, s is an integer satisfying 1 ≤ s < r, and ϵ ∈ {±1}, (2)

αg(p) = α(p) + 1 and αf(p) = λα(p).

We start with a lemma describing which values of f′(q) and g′(q) can occur.

Lemma 3.2. Assume that one of assumptions A.-D. holds. Then g′(q) = 1 , and either

(1)

g^{(i)}(q) = 0 for 2 ≤ i ≤ r

(in particular, neither assumption A. nor D. can hold in this case), or (2) f′(q) = (\frac{1}{λ})^{\frac{1}{s}} for some integer 1 ≤ s < r, and

g^{(i)}(q) = 0 for 2 ≤ i ≤ s, and g^{(s+1)}(q) ≠ 0.

Proof: Since fg = g^{λ}f,

f′(g(p)) g′(p) = (g^{λ})′(f(p)) f′(p).

When p = q, we thus have g′(q) = (g^{λ})′(q). But (g^{λ})′(q) = (g′(q))^{λ}, and so g′(q) = 1.

Suppose that f′(q) = κ ≠ 1. Then there is an interval [q, p) on which f is C^{r} conjugate to the linear map x ↦ κx ([St], Theorem 2). So in local coordinates, identifying q with 0,

f(x) = κx, and
g(x) = x + ax^{s+1} + o(x^{s+1}) for some s ≥ 1.

Then

fgf^{−1}(x) = x + (\frac{a}{κ^{s}})x^{s+1} + o(x^{s+1}), and
g^{λ}(x) = x + λax^{s+1} + o(x^{s+1}).

So either (1) a = 0, and therefore g^{(i)}(q) = 0 for 2 ≤ i ≤ r, or

(2) a ≠ 0, in which case κ = (\frac{1}{λ})^{\frac{1}{s}}, and

g^{(i)}(q) = 0 for 2 ≤ i ≤ s
≠ 0 for i = s + 1.

Now suppose that f′(q) = 1. Then in a neighborhood of q, we can write

f(x) = x + bx^{k+1} + o(x^{k+1}) and g(x) = x + ax^{s+1} + o(x^{s+1})

for some k, s ≥ 1. If k = s, then

[f, g](x) := fgf^{−1}g^{−1}(x) = x + o(x^{s+1})
= g^{λ−1}(x) = x + (λ − 1)ax^{s+1} + o(x^{s+1}).

So a = 0, and hence g^{(i)}(0) = 0 for 2 ≤ i ≤ r.

If k ≠ s, then we use the following well known result (see, e.g., [RS]):

Lemma 3.3. If f(x) = x+bx^{k+1}+o(x^{k+1}) and g(x) = x+ax^{s+1}+o(x^{s+1}), and if s > k ≥ 1, then

[f, g](x) = x + (s − k)abx^{s+k} + o(x^{s+k}).

Assume that s > k. (If s < k, then the proof is similar). It follows from Lemma 3.3 that

x + (s − k)abx^{s+k} + o(x^{s+k}) = [f, g](x)
= g^{λ−1}(x) = x + (λ − 1)ax^{s+1} + o(x^{s+1}).

So either (1) k ≥ 2, and therefore a = 0 and g^{(i)}(q) = 0, for 2 ≤ i ≤ r, or

(2) k = 1, and therefore b = \frac{λ−1}{s−1}.

But if k = 1, then

x + (s − 1)2abx^{s+1} + o(x^{s+1}) = [f^{2}, g](x)
= f^{2}gf^{−2}g^{−1}(x) = g^{λ^{2}−1}(x)
= x + (λ^{2} − 1)ax^{s+1} + o(x^{s+1}).

So b = \frac{λ^{2}−1}{2(s−1)} = \frac{λ−1}{s−1}, which is impossible, since λ ≠ 1. Therefore g^{(i)}(q) = 0, for 2 ≤ i ≤ r. □

Lemma 3.4. Assume that A. B. C. or D. holds. Then there is a neighborhood [q, p_{1}) ⊂ [q, q_{1}) and a C^{r} map A : [q, p_{1}) → ℝ such that for all p ∈ [q, p_{1}),

(1) A(p) = oH(p)^{s}, where H : [q, p_{1}) → [0, ∞) is a C^{r} diffeomorphism, 1 ≤ s < r is an integer, and o ∈ {±1};

(2)

Af(p) = \frac{1}{λ}A(p);

(3)

Ag(p) = \frac{A(p)}{1 − A(p)}.

Consequently, f′(q) = (\frac{1}{λ})^{\frac{1}{s}} for some integer 1 ≤ s < r, and g^{(s+1)}(q) ≠ 0.

Before giving a proof of Lemma 3.4, we will show how this lemma implies Proposition 3.1.

Lemma 3.5. Assume that A. B. C. or D. holds. Let A, H, s, and o be given by Lemma 3.4. For p ∈ [q, p_{1}), let

h(p) = \frac{−1}{H(p)}; α_{0}(p) = \frac{−1}{A(p)}.

Then α_{0} extends to a C^{r} map α : (q, q_{1}) → (−∞, ∞) satisfying the conclusions of Proposition 3.1.

Proof: Lemma 3.4 implies that for all p ∈ (q, p_{1}), α_{0}g(p) = α_{0}(p)+1 and α_{0}f(p) = λα_{0}(p). Since α_{0} has been defined in a fundamental domain for g, we can now extend this map to a C^{r} diffeomorphism α from (q, q_{1}) to (−∞, ∞) as follows. Since g has no fixed points in (q, q_{1}), given any p ∈ (q, q_{1}), there is some j ∈ ℤ such that g^{j}(p) ∈ (q, p_{1}). Let α(p) = α_{0}(g^{j}(p)) − j (which is easily seen to be independent of choice of j). By construction, αg(p) = α(p) + 1 for all p ∈ (q, q_{1}). Since f(p) = (g^{−j})^{λ}fg^{j}(p), we also have:

αf(p) = (α(g^{−j})^{λ}α_{0}^{−1})(α_{0}fα_{0}^{−1})(α_{0}g^{j}(p)) = λ(α(p) + j) − jλ = λα(p).

Hence the conclusions of Proposition 3.1 hold. □

Proof of Lemma 3.4: We say that a C^{2} function c : [a, b) → [a, b) is a C^{2} contraction if c′ is positive on [a, b) and c(x) < x, for all x ∈ (a, b). Since g has no fixed points in (q, q_{1}), either g or g^{−1} is a C^{2} contraction. We will assume until the end of the proof that g is a C^{2} contraction. Replacing g by g^{−1} does not change the relation fgf^{−1} = g^{λ}.

Since g has no fixed points in (q, q_{1}), there is a a unique C^{1} vector field X_{0} on [q, q_{1}) that generates a C^{1} flow g^{t} such that g|_{[q,q_{1})} = g^{1} (Szekeres, see [Nav] for a discussion).

Lemma 3.6. For all j ∈ ℕ and x ∈ [q, q_{1}), f^{−j}gf^{j}(x) = g^{\frac{1}{λ^{j}}} (x).

Proof: We will use the following result of Kopell:

Lemma 3.7. ([Ko] Lemma 1) Let g ∈ Diff ^{2}[q, q_{1}) be a C^{2} contraction that embeds in a C^{1} flow g^{t}, so that g = g^{1}. If h ∈ Diff ^{1}[q, q_{1}) satisfies hg = gh, then h = g^{t} for some t ∈ ℝ.

It follows from the relation f^{j}gf^{−j} = g^{λ^{j}} that f^{−j}gf^{j} commutes with g, and therefore Lemma 3.7 implies that f^{−j}gf^{j} = g^{t} for some t ∈ ℝ. This relation also implies that (f^{−j}gf^{j})^{λ^{j}} = g. So f^{−j}gf^{j} = g^{\frac{1}{λ^{j}}}. □

Let κ = f′(q). We may assume, by Lemma 3.2, that κ ≠ 1, and therefore there is an interval (q, p_{1}) on which f has no fixed points, and a C^{r} diffeomorphism H : [q, p_{1}) → [0, ∞) such that H f H^{−1}(x) = κx ([St], Theorem 2). The diffeomorphism H is unique up to multiplication by a constant. Let F = HfH^{−1} and let G = HgH^{−1}. Since we have assumed that g is a contraction, we have g([q, p_{1})) ⊆ [q, p_{1}).

Let X be the push-forward of the vector field X_{0} to [0, ∞) under H, and let G^{t} be the semiflow generated by X_{0}, so that G = G^{1}.

Lemma 3.8. If F′(0) ≤ (\frac{1}{λ})^{\frac{1}{r−1}} and G is r-flat at 0, then X(x) = 0 on [0, ∞).

Proof: We will show that for all x ∈ [0, ∞),

lim_{t→0} \frac{G^{t}(x) − x}{t} = 0.

Since the limit exists, it is enough to show that it converges to 0 for a subsequence t_{i} → 0. We will use the subsequence t_{i} = \frac{1}{λ^{i}}. Writing κ = F′(0) as before, we have

F (x) = κx and G(x) = x + R(x),

where R(x)/x^{r} → 0 as x → 0, and therefore:

G^{t_{i}}(x) = F^{−i}GF^{i}(x) = x + \frac{1}{κ^{i}} R((κ^{i}x)).

So

0 ≤ lim_{i→∞} \frac{|G^{t_{i}}(x) − x|}{t_{i}} = lim_{i→∞} λ^{i} (\frac{1}{κ^{i}} |R(κ^{i}x)|)
= lim_{i→∞}(λκ^{r−1})^{i}x^{r} \frac{|R((κ^{i}x))|}{(κ^{i}x)^{r}}
≤ lim_{i→∞} x^{r} \frac{|R((κ^{i}x))|}{(κ^{i}x)^{r}} = 0,

since κ^{r−1} ≤ \frac{1}{λ}. □

Corollary 3.9. Under any assumption A.-D., g is not r-flat at q, and therefore f′(q) = (\frac{1}{λ})^{\frac{1}{s}}, for some integer 1 ≤ s < r.

Proof: Clearly g cannot be infinitely flat if A. or D. holds. Under assumption C., f′(q) < (\frac{1}{λ})^{\frac{1}{k}}, for some k > 0 and f, g are C^{k}, so C. reduces to B. By Lemma 3.8, under assumption B., if g is r-flat at q, then the semiflow G^{t} is tangent to the trivial vector field, X(x) = 0. But then G = id, and therefore g = id on [q, p_{1}), contradicting the assumption that g has no fixed points in (q, q_{1}). □

Lemma 3.10. If F′(0) = (\frac{1}{λ})^{\frac{1}{s}} for some integer 1 ≤ s < r, then for some a < 0, X(x) = ax^{s+1} on [0, ∞).

Proof: As in the proof of Lemma 3.8, it is enough to show that for all x ∈ [0, ∞), and for t_{i} = \frac{1}{λ^{i}},

lim_{i→∞} \frac{G^{t_{i}}(x) − x}{t_{i}} = ax^{s+1}

for some a ∈ ℝ. If F′(0) = (\frac{1}{λ})^{\frac{1}{s}} for some integer 1 ≤ s < r, then by Lemma 3.2,

F (x) = (\frac{1}{λ})^{\frac{1}{s}} x and G(x) = x + ax^{s+1} + R(x)

for some a ∈ ℝ, where R(x)/x^{s+1} → 0 as x → 0. The value of a depends on the choice of linearizing map h for f|_{[q,p_{1})}. For all i ∈ ℕ,

G^{t_{i}}(x) = F^{−i}GF^{i}(x) = λ^{\frac{i}{s}} G (\frac{x}{λ^{\frac{i}{s}}} )
= x + ax^{s+1} \frac{1}{λ^{i}} + λ^{\frac{i}{s}} R (\frac{x}{λ^{\frac{i}{s}}} ) .

So

lim_{i→∞} \frac{G^{t_{i}}(x) − x}{t_{i}} = lim_{i→∞} λ^{i} (ax^{s+1} \frac{1}{λ^{i}} + λ^{\frac{i}{s}} R (\frac{x}{λ^{\frac{i}{s}}} ))

= lim_{i→∞} ax^{s+1} + R (\frac{x}{λ^{\frac{i}{s}}} ) \frac{λ^{\frac{i(s+1)}{s}}}{x^{s+1}} x^{s+1} = ax^{s+1}

Since G is a contraction on [0, ∞), it follows that a ≤ 0, and since G ≠ id, we must have a < 0. □

Corollary 3.11. If one of A.-D. holds, then f′(q) = (\frac{1}{λ})^{\frac{1}{s}} for some integer 1 ≤ s < r, and, after a suitable rescaling of the linearizing map H,

G(x) = \frac{x}{(1 + x^{s})^{\frac{1}{s}}} .

Proof: Choose H so that a = −1/s. Solving the differential equation \frac{δ}{δt}G^{t}(x) = aG^{t}(x)^{s+1} with initial condition G^{0}(x) = x, we obtain G^{t}(x) = x/(1 + tx^{s})^{\frac{1}{s}}. Since G(x) = G^{1}(x), the conclusion follows. □

To complete the proof of Lemma 3.4 assuming that g is a contraction, let s, H be given by Corollary 3.11, and let o = −1. Then Corollary 3.11 implies that A(p) = o(H(p)^{s}) satisfies the desired conditions. If g is not a contraction, we replace g by g^{−1} in the proof. Setting o = 1, we obtain the desired conclusions. □

3.1. Idea behind an alternate proof of Proposition 3.1. Suppose that f and g ≠ id are C^{r} diffeomorphisms, defined in a neighborhood of 0 in ℝ, both fixing the origin, and satisfying the relation:

fgf^{−1} = g^{λ},

for some λ > 1. In this context, the conclusion of Proposition 3.1 can be reformulated as follows: f and g are conjugate, via −1/h, to the maps

x ↦ (\frac{1}{λ})^{\frac{1}{s}} x and x ↦ \frac{x}{(1 − ox^{s})^{\frac{1}{s}}}

for some integer 1 ≤ s ≤ r and some o ∈ {±1}. The proof of Proposition 3.1 uses vector fields; here we sketch an alternative proof of this reformulation, using the Schwarzian derivative. This sketch can be made into a complete proof of Proposition 3.1 under assumptions A. C. and D., but gives a weaker result in case B: for this proof we will need both r ≥ 2s+1 and f′(q) ≤ (\frac{1}{λ})^{1/(s−1)}, for some s ≥ 1.

For simplicity, assume that r = ω and that λ = 2. First note that, since g is not infinitely flat, Lemma 3.2 implies that f′(0) ∈ \{(\frac{1}{2})^{\frac{1}{s}} | s ≥ 1\}. After conjugating f and g by an analytic diffeomorphism, we may assume, then, that:

f(x) = \frac{x}{2^{\frac{1}{s}}} ,

for some s ≥ 1.

Let F (x) = f(x^{\frac{1}{s}})^{s} = x/2 and let G(x) = g(x^{\frac{1}{s}})^{s}. Rewriting the relation F GF^{−1} = G^{2}, we obtain:

\frac{1}{2}G(2x) = G^{2}(x);

rearranging and iterating this relation, we obtain:

G(x) = 2^{k}G^{2^{k}} (\frac{x}{2^{k}} ) , (5)

for all k ≥ 1.

Recall that the Schwarzian derivative of a C^{3} function H is defined by:

S(H)(x) = \frac{H′′′(x)}{H′(x)} − \frac{3}{2} (\frac{H′′(x)}{H′(x)})^{2} ,

and has the following properties:

(1) S(H)(x) = 0 for all x iff G is Möbius, and

(2) for any C^{3} function K, S(H ∘ K)(x) = K′(x)^{2}S(H)(K(x)) + S(K)(x). Combining these properties with (5), we will show that S(G) = 0, which implies that G is Möbius. Lemma 3.2 implies that G′(0) = 1 and G′′(0) ≠ 0, so we have G(x) = \frac{x}{1−ox} for o ∈ {±1}. Writing g(x) = G(x^{s})^{1/s}, we obtain the desired result.

The first thing to check is that G is C^{3}. To obtain this, we use a slightly stronger version of Lemma 3.2 (whose proof is left as an exercise), which states that, if g is not infinitely flat, then

g(x) = x + ax^{s+1} + bx^{2s+1} + ⋯

Performing the substitution G = g(x^{1/s})^{s} in this series, one finds that G is C^{3}. (This requires that g be at least C^{2s+1}, in contrast to the proof of Proposition 3.1, which requires only C^{s+1}).

Equation (5) implies that

S(G)(x) = \frac{1}{2^{2k}} S(G^{2^{k}})(\frac{x}{2^{k}}),

for all k ≥ 1. Thus, by the cocycle condition (2) of the Schwarzian, we have:

(6)  S(G)(x) = \frac{1}{2^{2k}} ∑_{i=1}^{2^{k}} S(G)(G^{i−1}(\frac{x}{2^{k}})) ((G^{i−1})′(\frac{x}{2^{k}}))^{2}

(7)  = \frac{1}{2^{2k}} ∑_{i=1}^{2^{k}} S(G)(x_{i}) (Π_{j=1}^{i−1}G′(x_{j}))^{2}

where x_{i} := G^{i−1}(\frac{x}{2^{k}}).

Fix x, and assume without loss of generality that G^{j}(x) → 0 as j → ∞. Since G is C^{3} and G′(0) = 1, there is a constant C > 0 such that |G′(x_{i})| ≤ 1 + \frac{C}{2^{k}}, and |S(G)(x_{i})| ≤ C, for all i between 1 and 2^{k} and all k ≥ 1. Combined with (6), this gives us a bound on the Schwarzian of G at x:

|S(G)(x)| ≤ \frac{C}{2^{2k}} ∑_{i=1}^{2^{k}} (1 + \frac{C}{2^{k}} )^{2(i−1)}

≤ \frac{C}{2^{2k}} (\frac{1 − (1 + \frac{C}{2^{k}})^{2^{k+1}}}{1 − (1 + \frac{C}{2^{k}})^{2}} )

≤ \frac{1}{2^{k}} (\frac{e^{2C} − 1}{2 + \frac{C}{2^{k}}}) ,

for all k ≥ 1. Hence S(G)(x) = 0, for all x, which implies that G is Möbius.

4. Further properties of ramified covers: proofs of Proposition 2.3, Lemma 2.4 and Proposition 2.5.

The next lemma describes a useful normal form for ramified covering maps.

Lemma 4.1. Let π : ℝP^{1} → ℝP^{1} be a ramified covering map over 0, where π^{−1}(0) = {x_{1}, …, x_{d}}. Let s = (s(x_{1}), …, s(x_{d}), o_{1}, …, o_{d}) be the signature of π. Then given any x_{i} ∈ π^{−1}(0), there is a neighborhood U of x_{i} and an analytic diffeomorphism h : U → ℝ such that for all x ∈ U,

π(x) = h(x)^{s(x_{i})}.

Proof: In local coordinates at x_{i}, identifying x_{i} with 0, we can write

π(x) = ax^{s}(1 + O(x))
= ax^{s}g(x)

where a > 0, s = s(x_{i}) and g(x) = (1 + O(x)). Let h(x) = a^{\frac{1}{s}} xg(x)^{\frac{1}{s}}. Then h(x) is analytic in a neighborhood of 0, and π(x) = h(x)^{s}. □

This lemma motivates the following definition.

Definition: A C^{r} ramified cover over p ∈ S^{1} is a map π : S^{1} → S^{1} satisfying: (1) π^{−1}(p) = {q_{1}, q_{2}, …, q_{d}}, where q_{1} < q_{2} < … < q_{d};

(2) the restriction of π to π^{−1}(S^{1} \ {p}) is a regular C^{r} covering map onto S^{1} \ {p} of degree d ≥ 1;

(3) for all 1 ≤ i ≤ d, there are neighborhoods U_{i} of q_{i} and V of p, and C^{r} charts h_{i} : U_{i} → ℝ and k_{i} : V → ℝ with h_{i}(q_{i}) = 0 and k_{i}(p) = 0, such that

k_{i} π h_{i}^{−1}(x) = x^{s_{i}}

for some integer s_{i} > 0.

Remark: By Lemma 4.1, a ramified cover is a C^{ω} ramified cover.

We define the signature of a C^{r} ramified cover in the obvious way.

Definition: Let π_{1} and π_{2} be C^{r} ramified covering maps of degree d over p_{1} and p_{2}, respectively. Fix an orientation preserving identification between π_{1}^{−1}(p) and π_{2}^{−1}(p) and between E(π_{1}) and E(π_{2}). Suppose that f ∈ Diff ^{r}(S^{1}) satisfies f(p_{1}) = p_{2}, and let ζ ∈ D_{d}. We say that f̂ ∈ Diff ^{r}(S^{1}) is a (π_{1}, π_{2}, ζ)-ramified lift of f if:

(1) f̂(q) = ζ(q), for all q ∈ π_{1}^{−1}(p_{1}),

(2) f̂(e) = ζ(e), for all e ∈ E(π_{1}), and

(3) the following diagram commutes:

[figure: S^{1}; S^{1}; S^{1}; S^{1}; f̂; f; π_{1}; π_{2}]

Lemma 4.2. Let π_{1}, π_{2} and f be as above. Suppose that ζ ∈ D_{d} satisfies

• ζ(s(π_{1})) = s(π_{2}), if f ∈ Diff _{+}^{r}(S^{1}), or
• ζ(s(π_{1})) = I(s(π_{2})), if f ∈ Diff _{−}^{r}(S^{1}),

where I : S_{d} → S_{d} is the involution that reverses the sign of the last d coordinates.

Then there exists a unique (π_{1}, π_{2}, ζ)-ramified lift of f. We denote this lift by f̂(π_{1}, π_{2}, ζ), or by f̂(π, ζ), if π_{1} = π_{2} = π.

Furthermore, we have that if ζ ∈ C_{d}, then f̂(π_{1}, π_{2}, ζ) ∈ Diff _{+}^{r}(S^{1}).

Proof: Suppose first that f preserves orientation. Since the restriction of π_{1} to π_{1}^{−1}(S^{1} \ {p_{1}}) and the restriction of π_{2} to π_{2}^{−1}(S^{1}\{p_{2}}) are both regular C^{r} covering maps of degree d, for any ζ ∈ D_{d} there is a unique C^{r} diffeomorphism f̂_{0} : S^{1}\π_{1}^{−1}{p_{1}} → S^{1}\π_{2}^{−1}{p_{2}} such that f̂_{0}(e) = ζ(e) for all e ∈ E(π_{1}), and the diagram (3) commutes on the restricted domains. The condition s(π_{2}) = ζ(s(π_{1})) implies that f̂_{0} extends to a unique homeomorphism f̂ such that f̂(q) = ζ(q), for all q ∈ π_{1}^{−1}(p_{1}) and such that the diagram in (3) commutes. It remains to show that f̂ is a C^{r} diffeomorphism.

It suffices to show that f̂ is a C^{r} diffeomorphism at each q ∈ π_{1}^{−1}(p_{1}). By Lemma 5.6, there are local coordinates near q and f̂(q), identifying both of these points with 0, such that

(f̂(x))^{s(f̂(q))} = f(x^{s(q)})

for some integers s(f̂(q)) and s(q). Since s(π_{2}) = ζ(s(π_{1})), we have s(ζ(q)) = s(q). Let j = s(q) = s(f̂(q)). Since f is C^{r} and has a fixed point at 0,

f(x) = a_{1}x + a_{2}x^{2} + … + x^{r} + o(x^{r})

and we can assume that the coordinates have been chosen so that a_{1} > 0. So near x = 0,

f̂(x) = (a_{1}x^{j} + a_{2}x^{2j} + … + x^{rj} + o(x^{rj}))^{\frac{1}{j}}
= x (a_{1} + a_{2}x^{j} + … + x^{(r−1)j} + o(x^{(r−1)j}))^{\frac{1}{j}} ,

where the root is chosen so that f̂′(0) > 0. Since a_{1} > 0 and r ≥ 2, f̂ is a C^{r} diffeomorphism at 0. Similarly, we see that if f is analytic, then f̂ is analytic. Finally, we note that since f is orientation preserving, if ζ ∈ C_{d}, then f̂ must also be orientation preserving.

Now suppose that f ∈ Diff_{−}^{r}(S^{1}), and that ζ(s(π_{1})) = I(s(π_{2})). Let \overline{π}_{1} = f ∘ π_{1}. Setting f̂ to be the (\overline{π}_{1}, π_{2}, ζ)-lift of the identity map, we obtain the desired conclusions. □

Lemma 4.3. Let f_{1} and f_{2} be C^{r} diffeomorphisms of S^{1}, both with a fixed point at p, let π : S^{1} → S^{1} be a C^{r} ramified covering map over p with signature s, and let ζ_{1}, ζ_{2} ∈ D_{d}. Suppose that ζ_{1} and ζ_{2} satisfy ζ_{i}(s) = s if f_{i} ∈ Diff _{+}^{r}(S^{1}), and ζ_{i}(s) = I(s) if f_{i} ∈ Diff _{−}^{r}(S^{1}). Then

f̂_{2}(π, ζ_{2}) ∘ f̂_{1}(π, ζ_{1}) = { \atop f_{2} ∘ f_{1}}(π, ζ_{2} ∘ ζ_{1}).

Proof: The map f̂_{2}(π, ζ_{2}) ∘ f̂_{1}(π, ζ_{1})(q) satisfies:

(1) f̂_{2}(π, ζ_{2}) ∘ f̂_{1}(π, ζ_{1})(q) = ζ_{2} ∘ ζ_{1}(q), for all q ∈ π^{−1}(p),

(2) f̂_{2}(π, ζ_{2}) ∘ f̂_{1}(π, ζ_{1})(e) = ζ_{2} ∘ ζ_{1}(e), for all e ∈ E(π), and

(3) the following diagram commutes:

[figure: S^{1}; S^{1}; S^{1}; S^{1}; S^{1}; S^{1}; f̂_{1}(π, ζ_{1}); f_{1}; π; π; f̂_{2}(π, ζ_{2}); f_{2}; π]

By Lemma 4.2, we must have f̂_{2}(π, ζ_{2}) ∘ f̂_{1}(π, ζ_{1}) = { \atop f_{2} ∘ f_{1}}(π, ζ_{2} ∘ ζ_{1}). □

The following proposition is a C^{r} version of Proposition 2.3. Proposition 4.4. Suppose that G is a group, and that ρ : G → Diff _{+}^{r}(S^{1}) is a representation with global fixed point p. Let π : S^{1} → S^{1} be a C^{r} ramified cover over p with signature vector s. Then for every homomorphism h : G → Stab_{D_{d}}(s), there is a unique representation

ρ̂ = ρ̂(π, h) : G → Diff ^{r}(S^{1})

such that, for all γ ∈ G, ρ̂(γ) is the (π, h(γ))- ramified lift of ρ(γ). If h takes values in Stab_{C_{d}}(s), then ρ̂ takes values in Diff _{+}^{r}(S^{1}).

Proof: This follows immediately from the previous two lemmas. □

The following lemma is a C^{r} version of Lemma 2.4. Lemma 4.5. Let G be a group, and let ρ : G → Diff ^{ω}(S^{1}) be a representation with global fixed point p. Let π_{1}, π_{2} : S^{1} → S^{1} be C^{r} ramified covers over p ∈ S^{1}, with s(π_{1}) = ζ(s(π_{2})), for some ζ ∈ D_{d}.

Then for every homomorphism h : G → Stab_{D_{d}}(s), the representation ρ̃(π_{1}, h) is conjugate to ρ̃(π_{2}, ζhζ^{−1}) in Diff ^{r}(S^{1}), where (ζhζ^{−1})(γ) := ζh(γ)ζ^{−1}. If ρ takes values in Diff _{+}^{ω}(S^{1}), if ζ ∈ C_{d}, and if h takes values in Stab_{C_{d}}(s), then ρ̃(π_{1}, h) and ρ̃(π_{2}, ζhζ^{−1}) are conjugate in Diff _{+}^{r}(S^{1}).

Proof: This lemma follows from the diagram below, which commutes by Proposition 4.4 and Lemma 4.2. (Here îd = îd(π_{1}, π_{2}, ζ^{−1})).

[figure: S^{1}; S^{1}; S^{1}; S^{1}; S^{1}; S^{1}; S^{1}; S^{1}; ρ̂(π_{1}, h); ρ; π_{1}; π_{1}; ρ̂(π_{2}, ζhζ^{−1}); ρ; π_{2}; π_{2}; îd; îd; id; id]

□

Consider two lifts f̂(π_{1}, ζ), f̂(π_{2}, ζ) of the same diffeomorphism f (or, more generally, of conjugate diffeomorphisms). For purely topological reasons, if these lifts are conjugate by a map with rotation number 0, then s(π_{1}) and s(π_{2}) have the same length 2d, and the final d entries in these vectors must agree. (More generally, if the conjugacy has nonzero rotation number, then the final d entries of the first vector must lie in the D_{d}-orbit of the final d entries of the second). We now examine the first d entries of both vectors. We show that, under appropriate regularity assumptions on f and on the conjugacy, these entries must also agree, so that s(π_{1}) = s(π_{2}). The next lemma is the key reason for this.

Lemma 4.6. Let c : [0, ∞) → [0, ∞) be a C^{2} contraction. Suppose that, for some integers m, n > 0, the maps v_{1}(x) = c(x^{m})^{1/m} and v_{2}(x) = c(x^{n})^{1/n} are conjugate by a C^{1} diffeomorphism h : [0, ∞) → [0, ∞). Then m = n.

Proof: Since c is a C^{2} contraction, the standard distortion estimate (see, e.g. [Ko]) implies that for all x, y ∈ [0, ∞), there exists an M ≥ 1, such that for all k ≥ 0,

(8)  \frac{1}{M} ≤ \frac{(c^{k})′(x)}{(c^{k})′(y)} ≤ M.

Assume without loss of generality that n > m and suppose that there exists a C^{1} diffeomorphism h : [0, ∞) → [0, ∞) such that hv_{1}(x) = v_{2}(h(x)), for all x ∈ [0, ∞). Let H(x) = h(x^{1/m})^{n}. Note that the C^{1} function H : [0, ∞) → [0, ∞) has the following properties:

(1) H′(x) ≥ 0, for all x ∈ [0, ∞), and H′(x) = 0 iff x = 0;

(2) for all k ≥ 0, H ∘ c^{k} = c^{k} ∘ H. Then (2) implies that for every x ∈ [0, ∞):

H′(x) = H′(c^{k}(x))\frac{(c^{k})′(x)}{(c^{k})′(H(x))}

for all k ≥ 0. But (8) implies that (c^{k})′(x)/(c^{k})′(H(x)) is bounded independently of k, so that H′(x) = lim_{k→∞} H′(c^{k}(x)) = 0, contradicting property (1). □

Corollary 4.7. Let G and H be infinite subgroups of Âff^{s_{1}}(ℝ) and Âff^{s_{2}}(ℝ), respectively, for some s_{1}, s_{2} ∈ \overline{S}. If there exists α ∈ Diff ^{1}(S^{1}) such that αGα^{−1} = H, then s_{1} = s_{2}.

If G < Âff_{+}^{s_{1}}(ℝ) and H < Âff_{+}^{s_{2}}(ℝ), with s_{1}, s_{2} ∈ \overline{S}_{+}, and there exists α ∈ Diff _{+}^{1}(S^{1}) such that αGα^{−1} = H, then s_{1} = s_{2}.

Proof: Let G, H and α be given. Note that s_{1} and s_{2} must have the same length 2d, since the global finite invariant sets of G and H must be isomorphic. Let g, h be elements of G and H with rotation number 0 such that h = αgα^{−1}. Since dilations have twice as many fixed points in ℝP^{1} as translations, if g is a ramified lift of a translation, then so is h. Assume that g = Ŝ(π_{s_{1}}, id) and h = T̂(π_{s_{2}}, id), where S : x ↦ x + s and T : x ↦ x + t are translations with s, t > 0. Let q_{1}, …, q_{d} and α(q_{1}), …, α(q_{d}) be the preimages of ∞ under π_{s_{1}} and π_{s_{2}}, respectively. In a neighborhood of q_{i}, the map g is conjugate to x ↦ (S(x)^{m_{i}})^{1/m_{i}} and in a neighborhood of α(q_{i}), h is conjugate to x ↦ (T (x)^{n_{i}})^{1/n_{i}}, where m_{i} = s(q_{i}) and n_{i} = s(α(q_{i})). Since S is a C^{2} contraction in a neighborhood of ∞ and T is conjugate to S, it follows from Lemma 4.6 that m_{i} = n_{i} for 1 ≤ i ≤ d, which implies that s_{1} = s_{2}.

Suppose instead that g is a ramified lift of a map in Aff(ℝ) conjugate to the dilation D : x ↦ ax, for some a > 1. Since g must have d fixed points with derivative a, so must h, and so h is also a ramified lift of a map in Aff(ℝ) conjugate to D. Around ∞, the map D is a C^{2} contraction, and the same proof as above shows that s_{1} = s_{2}.

The proof in the orientation-preserving case is analogous. □

Proof of Proposition 2.5: Let ρ_{n} : BS(1, n) → Diff ^{ω}(S^{1}) be the standard representation. Suppose that ρ̂_{n}(π_{s_{1}}, h_{1}) and ρ̂_{n}(π_{s_{2}}, h_{2}) ∈ V are conjugate by α ∈ Diff ^{1}(S^{1}), where s_{1}, s_{2} ∈ S. It follows from Corollary 4.7 that s_{1} = s_{s}.

We next show h_{1} = h_{2}. Let γ ∈ BS(1, n) and let k = ρ_{n}(γ). Let k_{1} = k̂(π_{s_{1}}, h_{1}(γ)) and k_{2} = k̂(π_{s_{2}}, h_{2}(γ)) Then for all q ∈ π_{s_{1}}^{−1}(∞), we have:

αh_{1}(γ)(q) = αk_{1}(q) = k_{2}(α(q)) = h_{2}(γ)(α(q)),

and for all e ∈ E(π_{1}),

αh_{1}(γ)(e) = αk_{1}(e) = k_{2}(α(e)) = h_{2}(γ)(α(e)).

Since α(π_{s_{1}}^{−1}(∞)) = π_{s_{1}}^{−1}(∞) and α(E(π_{1})) = E(π_{1}), it follows that h_{1}(γ) = h_{2}(γ). So αh_{1} = h_{2}α. Recall that each element ρ̂_{n}(π_{s}, h) of V is given by a signature vector s ∈ \overline{S} and a representative h of a conjugacy class in Hom(BS(1, n), Stab_{D_{d}}(s)). So h_{1} = h_{2}. □

5. Proof of Theorems 1.1 and 1.3

The construction behind this proof is very simple. We are given a C^{r} representation ρ of BS(1, n). Using elementary arguments, we are reduced to the case where f = ρ(a) and g = ρ(b) have a common finite invariant set, the set of periodic orbits of g. Assume that the rotation numbers of f and g are both 0. Using the results from Section 3, we obtain a local characterization of f and g on the intervals between the common fixed points. On each of these intervals, f is conjugate to the dilation x ↦ nx and g is conjugate to the translation x ↦ x + 1. Gluing together the conjugating maps gives us a C^{r} ramified covering map over ∞. Hence ρ is a C^{r} ramified lift of the standard representation. Proposition 2.2 implies that there is a rational ramified cover with the same signature as the the given C^{r} ramified cover. Lemma 4.5 implies that ρ is C^{r} conjugate to a ramified lift of ρ_{n} under the rational ramified cover. It remains to handle the case where the rotation numbers of f and g are not 0, but this is fairly simple to do, since the elements of the standard representation embed in analytic vector fields. We now give the complete proof.

Let ρ : BS(1, n) → Diff ^{r}(S^{1}) be a representation, where r ∈ [2, ∞], or r = ω. If r < ∞,

assume that σ(ρ) ≤ (\frac{1}{n})^{\frac{1}{r−1}} . If r = ∞, we assume that σ(ρ) < 1.

Let f = ρ(a) and g = ρ(b), where aba^{−1} = b^{n}. Since g is conjugate to g^{n}, it follows that τ(g) = ±τ(g^{n}) = ±nτ(g), where τ(h) denotes the rotation number of h ∈ Homeo (S^{1}). Hence g has rational rotation number.

Lemma 5.1. f preserves the set of periodic points of g.

Proof: This follows from the relation fg = g^{n}f. If g^{k}(q) = q, then g^{nk}(f(q)) = fg^{k}(q) = f(q). So f(q) is also periodic for g. □

Suppose that τ(f) is irrational. Then by Lemma 5.1, the periodic points of g are dense in S^{1}, which implies that g^{k} = id, for some k ≤ n + 1. This implies that conclusion (1) of Theorem 1.1 holds.

Suppose, on the other hand, that τ(f) is rational. Choose l so that g^{l} and f^{l} are both orientation-preserving and both have rotation number 0. Then f^{l} leaves Fix(g^{l}) invariant. Choose p ∈ Fix(g^{l}). Any accumulation of {f^{ln}(p)} must be a fixed point for f^{l} and for g^{l}. We have shown:

Lemma 5.2. f^{l} and g^{l} have a common fixed point.

Note that the fixed points for f^{l} are isolated; if f is not analytic, then σ(ρ) < 1, which implies that the fixed points for f^{l} are hyperbolic. Let w_{1} < w_{2} < … < w_{k} be the set of fixed points of f^{l}. We will see that if g^{l} is not the identity map, then the set of fixed points for g^{l} is exactly equal to the set of sinks for f^{l}.

Lemma 5.3. If g^{l}(w_{i}) = w_{i} and (f^{l})′(w_{i}) > 1, then g^{l} = id on [w_{i−1}, w_{i+1}].

Proof: Suppose that (f^{l})′(w_{i}) = λ > 1, and let α : [w_{i}, w_{i+1}) → [0, ∞) be a C^{1} linearizing diffeomorphism such that αf^{l}α^{−1}(x) = λx for all x ∈ [0, ∞). Let F = αf^{l}α^{−1}, and let G = αg^{l}α^{−1}. If g^{l} ≠ id on [w_{i}, w_{i+1}), then there is a point x_{0} ∈ [0, ∞) such that G(x_{0}) ≠ x_{0}. Let x_{0} be any such point. We may assume that G^{k}(x_{0}) → c as k → ∞, for some c < ∞, because this will be true for either G or G^{−1}. Since GF^{−k} = F^{−k}G^{n^{k}} for all k ∈ ℕ, it follows that

G′(F^{−k}(x_{0})) = \frac{(F^{−k})′(G^{n^{k}}(x_{0}))}{(F^{−k})′(x_{0})} (G^{n^{k}})′(x_{0}),

for all k ∈ ℕ. But since G′(0) = 1 (by Lemma 3.2), this means that (G^{n^{k}})′(x_{0}) → 1, as k → ∞ (or k → −∞), for every point x_{0} that is not fixed by G. Since G is not the identity, this is not possible. Hence g = id on [w_{i}, w_{i+1}]. A similar argument shows that g = id on [w_{i−1}, w_{i}], □

Corollary 5.4. If g^{l} has a fixed point in the interval (w_{i}, w_{i+1}), then g^{l} = id on [w_{i}, w_{i+1}]. That is, ∂Fix(g^{l}) ⊆ Fix(f^{l}).

Proof: Suppose that g^{l}(p) = p for some p ∈ (w_{i}, w_{i+1}), and suppose that f^{kl}(p) → w_{i} as k → −∞. By Lemma 5.1, f^{lk}(p) is periodic for g^{l} for all k ∈ ℤ. Since g^{l} is an orientation preserving circle diffeomorphism with a fixed point, f^{lk}(p) is a fixed point of g^{l} for all k. By continuity, w_{i} is a common fixed point for f^{l} and g^{l}. Since (f^{l})′(w_{i}) > 1, Lemma 5.3 implies that g^{l} = id on [w_{i}, w_{i+1}]. Similarly, if f^{kl}(p) → w_{i+1} as k → −∞, then g = id on [w_{i}, w_{i+1}]. □

This has the immediate corollary:

Corollary 5.5. f^{l} fixes every component of S^{1} \ Fix(g^{l}).

Remark: Corollary 5.5 also follows from Theorem 1.6. We have given a different proof here since we will need Lemma 5.3 for the proof of Lemma 5.6.

Let −∞ ≤ q_{1} < q_{2} < ⋯ < q_{d} < ∞ be the elements of ∂Fix(g^{l}).

Lemma 5.6. On each interval (q_{i−1}, q_{i}], either g^{l} = id, or there is a C^{r} map α_{i} : (q_{i−1}, q_{i}] → (−∞, ∞] such that

(1) α_{i} conjugates f^{l} to the map x ↦ n^{l}x, and conjugates g^{l} to the map x ↦ x + 1;

(2) α_{i}|_{(q_{i−1},q_{i})} is a C^{r} diffeomorphism onto (−∞, ∞)

(3) For all p in a neighborhood of q_{i},

α_{i}(p) = o_{i}h(p)^{s}

where h is a C^{r} orientation-preserving diffeomorphism onto a neighborhood of ∞, 1 ≤ s < r, and o_{i} ∈ {±1}.

Proof: This follows from Proposition 3.1. Note that we can apply Proposition 3.1 in this setting since we know that if g^{l} ≠ id on (q_{i−1}, q_{i}], then (f^{l})′(q_{i}) ≤ 1 (by Lemma 5.3). By our assumptions on σ(ρ), if 2 ≤ r < ∞, then (f^{l})′(q_{i}) ≤ (\frac{1}{n})^{\frac{1}{r−1}} , and if r = ∞, then (f^{l})′(q_{i}) < 1. Therefore one of the assumptions A. - C. of Proposition 3.1 will hold. □

Corollary 5.7. Either g^{l} = id, or ∂Fix(g^{l}) = Fix(g^{l}) = {q_{1}, …, q_{d}}.

Proof: Assume that ∂Fix(g^{l}) = {q_{1}, …, q_{d}} ≠ Fix(g^{l}), but g^{l} ≠ id. Then there is an interval [q_{i−1}, q_{i}] on which g^{l} = id but where g^{l} ≠ id on [q_{i}, q_{i+1}]. By Lemma 3.4, either g^{l} or g^{−l} is C^{r} conjugate in a neighborhood [q_{i}, p) to the map x ↦ x/(1 − x^{s})^{\frac{1}{s}} for some integer 1 ≤ s < r. But this map is not r-flat at x = 0, so g^{l} is not C^{r} at q_{i}, a contradiction. □

Corollary 5.8. If g^{l} ≠ id, then the map π : S^{1} → ℝP^{1} defined by:

π(p) = α_{i}(p), for p ∈ (q_{i−1}, q_{i}]

is a C^{r} ramified covering map over ∞, f^{l} is a π-ramified lift of x ↦ n^{l}x, and g^{l} is a π-ramified lift of the map x ↦ x + 1.

Proof: Let q_{i} ∈ Fix(g^{l}). Applying Lemma 5.6 to the interval [q_{i}, q_{i+1}), we obtain a map \overline{α}_{i+1} : [q_{i}, q_{i+1}) → [−∞, ∞) which is a C^{r} diffeomorphism on (q_{i}, q_{i+1}), and which is a power of a C^{r} diffeomorphism in a (right) neighborhood of q_{i}; \overline{α}_{i+1}(p) = h(p)^{s} for p near q_{i}, for some C^{r} diffeomorphism h and some integer 1 ≤ s < r. Similarly, on the interval (q_{i−1}, q_{i}] there is a map α_{i} : (q_{i−1}, q_{i}] → (∞, −∞] which is a power of a diffeomorphism in a (left) neighborhood of q_{i}; α_{i}(p) = h_{∗}(p)^{s_{∗}} near q_{i}. We will show that s = s_{∗}, and that the diffeomorphisms h and h_{∗} glue together to give a C^{r} diffeomorphism in a neighborhood of q_{i}. This will prove that the map

π_{i}(p) = \begin{cases}α_{i}(p), for p ∈ (q_{i−1}, q_{i}] \\ \overline{α}_{i+1}(p), for p ∈ [q_{i}, q_{i+1}) \end{cases}

is the restriction to (q_{i−1}, q_{i+1}) of a C^{r} ramified covering map over ∞. By construction, the restrictions of f^{l} and g^{l} to (q_{i−1}, q_{i+1}) are π_{i}-ramified lifts of the maps x ↦ n^{l}x and x ↦ x + 1 respectively.

The diffeomorphism 1/h maps q_{i} to 0, and conjugates g^{l} to the map x ↦ x/(1 + x^{s})^{1/s}. Similarly, 1/h_{∗} conjugates g^{l} to x ↦ x/(1 + x^{s_{∗}})^{1/s_{∗}}. Since g is C^{r}, we must have s = s_{∗}. Both 1/h and 1/h_{∗} are linearizing maps for f^{l} at q_{i}, and it is not hard to see that they define a C^{r} diffeomorphism H in a neighborhood of q_{i}. Therefore h and h_{∗} glue together to give a C^{r} diffeomorphism 1/H in a neighborhood of q_{i}.

It remains to show that π_{i} = π_{i+1} on (q_{i}, q_{i+1}). Since the restriction of both of these maps to (q_{i}, q_{i+1}) are diffeomorphisms which linearize f^{l}, they are the same up to a constant multiple. There is a unique point x_{0} ∈ (q_{i−1}, q_{i}) satisfying f^{l}(x_{0}) = g^{n^{l}−l}(x_{0}); – this is the point x_{0} = g^{l}(y), where y is the unique fixed point for f^{l} in (q_{i}, q_{i+1}). Both π_{i} and π_{i+1} send the point x_{0} to the same point 1 ∈ ℝ. So we have π_{i} = π_{i+1} on (q_{i}, q_{i+1}). □

It follows from Lemma 4.5 that the representation of BS(1, n^{l}) generated by f^{l} and g^{l} is C^{r} conjugate to an element of V. In the remainder of this section, we will show that the diffeomorphisms f and g are C^{r} ramified lifts of the generators of the standard action of BS(1, n) on S^{1}, hence the representation they generate is also C^{r} conjugate to an element of V. We begin with some lemmas about ramified lifts of flows on S^{1}.

Lemma 5.9. Let φ : S^{1} → S^{1} be a C^{r} flow with a fixed point at p, and let π : S^{1} → S^{1} be a C^{r} ramified covering map over p. Let F = φ̂^{1}(π, id) be the π-ramified lift of the time-1 map φ^{1} with rotation number zero. Then F embeds as the time-1 map of a C^{r} flow F^{t} on S^{1}, and for all t ∈ ℝ, F^{t} = φ̂^{t}(π, id).

Proof: By Lemma 4.2, given any t ∈ ℝ there is a unique (π, id) - ramified lift of φ^{t}, F^{t} := φ̂^{t}(π, id). Lemma 4.3 imples that F^{t} ∘ F^{s} = F^{s+t} = F^{s} ∘ F^{t} for all s, t ∈ ℝ.

Let X be the C^{r−1} vector field that generates φ, and let X̂ be the lift of X under π. This vector field is clearly C^{r−1} on S^{1} \ π^{−1}(p) and clearly generates the flow F^{t} on S^{1}. In a neighborhood of q_{i} ∈ π^{−1}(p), X̂ takes the form

X̂(q) = d_{π(q)}π^{−1}X(πq),

and π takes the form π(x) = x^{s}. A straightforward calculation shows that the vector field X̂ is C^{r−1}. Similarly, X̂ is analytic if X and π are. This completes the proof. □

Lemma 5.10. Let F : S^{1} → S^{1} be the time-1 map of a C^{r} flow F^{t}, where r ≥ 2. Suppose that F is not r-flat, and τ(F) = 0. If G is a C^{r} orientation preserving diffeomorphism such that F G = GF, and if τ(G) = 0, then G = F^{t} for some t ∈ ℝ.

Proof: Since τ(F) = 0 and F is not r-flat, F has a finite set of fixed points. Let q_{1} < … < q_{d} be the elements of Fix(F). If F G = GF, then G permutes the fixed points of F, and since G is orientation preserving and has rotation number zero, G([q_{i}, q_{i+1}]) = [q_{i}, q_{i+1}] for all q_{i} ∈ Fix(F). By Lemma 3.7, on [q_{i}, q_{i+1}), G = F^{t_{i}} for some t_{i} ∈ ℝ, and on (q_{i}, q_{i+1}], G = F^{s_{i}} for some s_{i}. Clearly, t_{i} = s_{i}. So for 1 ≤ i ≤ d,

G|_{[q_{i},q_{i+1}]} = F^{t_{i}}, for some t_{i} ∈ ℝ.

If F′(q_{i}) ≠ 1 for some q_{i} ∈ Fix(F), then since G is C^{1} at q_{i}, it follows that t_{i} = t_{i−1}. If F′(q_{i}) = 1, then in local coordinates, identifying q_{i} with 0,

F (x) = F^{1}(x) = x + ax^{k} + o(x^{k})

for some a ≠ 0 and k ≤ r. Therefore

G(x) = \begin{cases}x + t_{i−1}ax^{k} + o(x^{k}), & for x ∈ (q_{i−1}, q_{i}] \\ x + t_{i}ax^{k} + o(x^{k}), & for x ∈ [q_{i}, q_{i+1}). \end{cases}

Since k ≤ r and G is C^{r}, t_{i} = t_{i−1}. □

Corollary 5.11. Let π : S^{1} → S^{1} be a C^{r} ramified covering map over ∞, and let F = k̂(π, id) be the (π, id)-ramified lift of k ∈ Aff(ℝ), k ≠ id. Let s(π) = (s_{1}, …, s_{d}, o_{1}, …, o_{d}), where s_{i} ≤ r − 1 for 1 ≤ i ≤ d. By Lemma 5.9, F embeds as the time-1 map of a C^{r} flow F^{t}. If H : S^{1} → S^{1} is a C^{r} orientation preserving diffeomorphism such that F H = HF, and if τ(H) = 0, then H = F^{t} for some t ∈ ℝ.

Proof: By Lemma 5.10, it is enough to show that F is not r-flat. In coordinates identifying a fixed point with 0,

F (x) = \frac{x}{(b + ax^{s})^{\frac{1}{s}}} ,

where either a ≠ 1 or b ≠ 0, which is clearly not r-flat, if s < r. □

Proposition 5.12. Let F ∈ Diff ^{r}(S^{1}) be a diffeomorphism such that F^{l} is orientationpreserving and τ(F^{l}) = 0, for some l > 0. Suppose that F^{l} = k̂^{l}(π, id) is a C^{r} ramified lift of k^{l} ≠ id, where k ∈ Aff_{+}(ℝ), and suppose that s(π) = (s(q_{1}), …, s(q_{d}), o_{1}, …, o_{d}), where s(q_{i}) ≤ r − 1 for 1 ≤ i ≤ d. Then either F is a π-ramified lift of k or F is a π-ramified lift of −k.

Proof: Let ζ ∈ D_{d} be such that ζ(q) = F (q) for all q ∈ π^{−1}(∞), and ζ(e) = F (e) for all e ∈ E(π).

Lemma 5.13. ζ ∈ Stab _{D_{d}}^{#}(s(π)).

Proof: Given any q ∈ π^{−1}(∞), there is an interval [q, p) and C^{r} diffeomorphisms h_{1} : [q, p) → [0, ∞) and h_{2} : [F (q), F (p)) → [0, ∞) such that

h_{1}F^{l}h_{1}^{−1}(x) = [k^{l}(x^{s})]^{\frac{1}{s}} , and h_{2}F^{l}h_{2}^{−1}(x) = [k^{l}(x^{t})]^{\frac{1}{t}} ,

where s = s(q) and t = s(F (q)). We can assume that k^{l} is a contraction on [0, ∞). (If not, then use k^{−l} and F^{−l}). Since F^{l}|_{[q,p)} is conjugate by F to F^{l}|_{[F (q),F (p))}, Lemma 4.6 implies that s(q) = s(F (q)), and therefore ζ ∈ Stab_{D_{d}}^{#}(s(π)). □

By Lemma 5.13, either ζ(s(π)) = s(π), or ζ(s(π)) = I(s(π)). If ζ(s(π)) = s(π), then let α : S^{1} → S^{1} be the (π, ζ)-ramified lift of the identity map: α = îd(π, ζ). By Lemma 4.3, α commutes with F^{l}. So F α^{−1} commutes with F^{l}, and by construction, F α^{−1} fixes every interval (q_{i}, q_{i+1}) ⊆ π^{−1}(ℝP^{1} \ {∞}). By Lemma 5.10, F^{l} embeds as the time-1 map of a C^{r} flow, F^{t}, and F α^{−1} = F^{t_{0}} = ϕ̂^{t_{0}}(π, id) for some t_{0} ∈ ℝ, where ϕ is an analytic flow with ϕ^{1} = k^{l}. Therefore F = ϕ̂^{t_{0}}(π, id) ∘ îd(π, ζ) = ϕ̂^{t_{0}}(π, ζ) (using Lemma 4.3). It follows that t_{0} = 1/l, and therefore F is the (π, ζ)-ramified lift of the map k.

If ζ(s(π)) = I(s(π)), then we let α = ˆ−id(π, ζ), the (π, ζ)-ramified lift of −id : x → −x. As above, F α^{−1} is the (π, ζ)-ramified lift of k, and therefore F = ψ^{t}α = ˆ−k(π, ζ). □

Corollary 5.14. f and g are C^{r} ramified lifts under π of the generators of the standard action of BS(1, n) on S^{1}.

Proof: The standard representation ρ_{n^{l}} : BS(1, n^{l}) → Diff ^{ω}(ℝP^{1}) is analytically conjugate to the representation κ : BS(1, n^{l}) → Diff ^{ω}(ℝP^{1}) with generators κ(a^{l}) : x ↦ n^{l}x and κ(b^{l}) : x ↦ x + l. So there is a C^{r} ramified covering map π : S^{1} → S^{1} over p such that f^{l} = { \atop κ(a^{l})}(π, id) and g^{l} = { \atop κ(b^{l})}(π, id). By Proposition 5.12, either f is a ramified lift of ρ_{n}(a) : x ↦ nx, or f is a ramified lift of −ρ_{n}(a) : x ↦ −nx. Similarly, g is either a ramified lift of ρ_{n}(b) : x ↦ x + 1, or a ramified lift of −ρ_{n}(b) : x ↦ −x − 1. Since f and g satisfy the relation fgf^{−1} = g^{n}, the maps that they are lifted from must also satisfy this relation. Given this requirement, the only possibility is that f is a π-ramified lift of ρ_{n}(a) and g is a π-ramified lift of ρ_{n}(b). □

Since the generators ρ(a) = f and ρ(b) = g of the representation ρ are ramified lifts under π of the generators ρ_{n}(a) and ρ_{n}(b), respectively, of ρ_{n}, it follows that, for every γ ∈ BS(1, n), there exists a unique h(γ) ∈ D_{d} (or in C_{d} if ρ is orientation-preserving) such that:

ρ(γ) = { \atop ρ_{n}(γ)}(π, h(γ)).

Since ρ(γ_{1}γ_{2}) = ρ(γ_{1})ρ(γ_{2}), it follows that h : BS(1, n) → D_{d} (C_{d}) is a homomorphism. Finally, note that h must take values in Stab_{D_{d}}(s) (or Stab_{C_{d}}(s), if ρ is orientation-preserving).

This concludes the proof of Theorems 1.1 and 1.3. □

Finally, we sketch the proof of Theorem 1.6.

Sketch of proof of Theorem 1.6. Let ρ be a C^{r} representation of BS(1, n), with r ≥ 2, let f = ρ(a) and g = ρ(b). We may assume that f has rational rotation number. By taking powers of the elements of BS(1, n), we may assume that both f and g have rotation number 0. Assume that g is not the identity map.

Let J be a component of the complement of Fix(g). Using a distortion estimate and the group relation one shows that J must be fixed by f, as follows. Otherwise, the f-orbit of J must accumulate at both ends on a fixed point of f. The standard C^{2} distortion estimate shows that there is an M > 1 such that for all x, y in same component of the f-orbit of J, and for all k ∈ ℤ,

\frac{1}{M} < |\frac{(f^{k})′(x)}{(f^{k})′(y)}| < M.

But, for all k ∈ N, we have that f^{k}gf^{−k} = g^{n^{k}}. Hence, for all p ∈ J, we have:

(g^{n^{k}})′(p) = g′(y)\frac{(f^{k})′(g(y))}{(f^{k})′(y)},

where y = f^{−k}(p). Note that y and g(y) lie in the same component f^{−k}(J), and g′(y) is uniformly bounded. This implies that for all p ∈ J and all k ∈ ℕ, (g^{n^{k}})′(p) is bounded, so that g = id on J, a contradiction.

So f fixes each component of the complement of Fix(g). Let J be such a component. Since g has no fixed points on J, g embeds in a C^{1} flow g^{t}, defined on J minus one of its endpoints, that is C^{r} in the interior of J (see, e.g. [ZL]). Furthermore, for all t, fg^{t}f^{−1} = g^{nt} (this follows from Kopell’s lemma). Fixing some point p in the interior of J, this flow defines a C^{r} diffeomorphism between the real line and the interior of J, sending t ∈ ℝ to g^{t}(p) ∈ J. Conjugating by this diffeomorphism, g^{t} is sent to a translation by t, and f is sent to a diffeomorphism F satisfying F (x + t) = F (x) + tn, for all t, x ∈ ℝ. But this means that F′(x) = n for all x ∈ ℝ. Up to an affine change of coordinates, g is conjugate on J to x ↦ x + 1 and f is conjugate to x ↦ nx. □

6. Proof of Proposition 1.5

Let ρ : BS(1, n) → Diff ^{ω}(S^{1}) be a π-ramified lift of the standard representation ρ_{n} with

σ(ρ) = (\frac{1}{n})^{\frac{1}{r−1}} , for some r ≥ 2. Let Q be the set of all points q ∈ π^{−1}(∞) satisfying

s(q) = r − 1; this set is nonempty since σ(ρ) = (\frac{1}{n})^{\frac{1}{r−1}} . Lemma 4.1 implies that, in a neighborhood of q, π : x ↦ x^{r−1}, in the appropriate coordinates identifying q with 0.

For t ∈ (−1, 1) we deform π to obtain a C^{r−1+t^{2}} map π_{t} : S^{1} → S^{1} with the following properties:

• π_{0} = π and π^{−1}(∞) = π_{t}^{−1}(∞), for all t;

• π_{t}|_{S^{1}\π_{t}^{−1}(∞)} is a C^{∞} covering map onto its image;

• about each q ∈ π^{−1}(∞) \ Q, π_{t} is locally equal to π;

• about each q ∈ Q, π_{t} is locally x ↦ x^{r−1+t^{2}}, in the same charts identifying q with 0 described above.

A slight modification of the proof of Proposition 4.4 also shows that ρ_{n} has a lift to a C^{r} representation ρ_{t} : BS(1, n) → Diff ^{r}(S^{1}) so that the following diagram commutes, for all γ ∈ BS(1, n):

[figure: S^{1}; S^{1}; S^{1}; S^{1}; ρ_{t}(γ); ρ_{n}(γ); π_{t}; π_{t}]

(One merely needs to check that the integer j in the proof of Lemma 4.2 can be replaced by the real number r − 1 + t^{2}).

Notice that ρ_{t} has the property that σ(ρ_{t}) = (\frac{1}{n})^{\frac{1}{r−1+t^{2}}} , so that ρ_{s} is not C^{1} conjugate to ρ_{t} unles s = t. One can further modify this construction by replacing the points of Q by intervals of length ε_{t}, extending ρ_{t}(b) isometrically across these intervals, and extending ρ_{t}(a) in an arbitrary C^{r} fashion to these intervals. Since ρ_{t}(b) is r-flat (by Lemma 3.2) on Q for t ≠ 0 and r −1 flat for t = 0, the representation ρ_{t} is C^{r} and varies C^{r−1} continuously in t if we choose ε_{t} → 0 as t → 0. In this way, one can create uncountably many deformations of ρ. (Note that, in essence, we have deformed π to obtain a “broken C^{r} ramified cover” à la Theorem 1.6). □

7. Proof of Theorem 1.9.

Let r ∈ {∞, ω}, and let G < Diff ^{r}(S^{1}) be a solvable group without infinitely flat elements. Suppose that G^{m} := {g^{m} : g ∈ G} is not abelian, for any m ∈ ℤ. We begin by showing that the group G^{2} < Diff_{+}^{r}(S^{1}) has a finite set of points that is globally invariant.

Lemma 7.1. G^{2} contains a non-trivial normal abelian subgroup N, such that N contains an element of infinite order. There is an integer d > 0, and a finite set {q_{1}, …, q_{d}}, with q_{1} < q_{2} < ⋯ < q_{d}, such that:

(1) for all f ∈ G^{2}, τ(f^{d}) = 0 and f{q_{1}, …, q_{d}} = {q_{1}, …, q_{d}};

(2) for all g ∈ N, either g^{d} = id or Fix(g^{d}) = {q_{1}, …, q_{d}}.

Proof: Note that G^{2} is a solvable group, and every diffeomorphism in G^{2} is orientation preserving. Let

G^{2} = G_{0} > G_{1} > … > G_{n} > G_{n+1} = {id}

be the derived series for G^{2}, and let N = G_{n} be the terminal subgroup in this series. Recall that N is a normal abelian subgroup of G^{2}. We first show that N contains an element of infinite order. We will use the following result of Ghys ([Gh2] Proposition 6.17):

Lemma 7.2. If H ⊂ Homeo_{+}(S^{1}) is solvable, then the rotation number τ : H → ℝ/ℤ is a homomorphism.

Suppose that every diffeomorphism in N has finite order. Since G^{2} ≠ N (because G^{2} is not abelian), G_{n−1} cannot be abelian – if it were, then G_{n} would be trivial. Suppose that f, h ∈ G_{n−1}. By Lemma 7.2, τ(fhf^{−1}h^{−1}) = 0. But fhf^{−1}h^{−1} is orientation preserving and has finite order, since fhf^{−1}h^{−1} ∈ G_{n}. Therefore fhf^{−1}h^{−1} = id, and G_{n−1} is abelian, a contradiction. So N contains a diffeomorphism with infinite order.

If τ(g) is irrational, for some orientation-preserving g ∈ N, then the elements of N are simultaneously conjugate to rotations. But, since N is normal in G^{2}, this implies that the elements of G^{2} are simultanously conjugate to rotations, which implies that G^{2} is abelian, a contradiction.

Hence τ(g) ∈ ℚ/ℤ, for every g ∈ N. Note that every g ∈ N either has finite order, or a finite set of periodic points: if Fix(g^{l}) is infinite, for some integer l ≠ 0, then there is a point q ∈ Fix(g^{l}) that is an accumulation point for a sequence {q_{i}} ⊂ Fix(g^{l}). But this implies that g^{l} is infinitely flat at q, and therefore g^{l} = id.

Hence there exists g ∈ N with infinite order and a finite fixed set, Fix(g) = {q_{1}, …, q_{d}}. If h ∈ N is another element of N, then, since h commutes with g, it follows that h({q_{1}, …, q_{d}}) = {q_{1}, …, q_{d}}, and so τ(h^{d}) = 0. If the set of fixed points for h^{d} is infinite, then h^{d} = id, and if Fix(h^{d}) is finite, then Fix(h^{d}) = {q_{1}, …, q_{d}}.

Finally, let f ∈ G^{2} \ N and pick g ∈ N satisfying Fix(g) = {q_{1}, …, q_{d}}. Then there exists a \overline{g} ∈ N such that fgf^{−1} = \overline{g}. This implies that f(Fix(g)) = Fix(\overline{g}); that is, f({q_{1}, …, q_{d}}) = {q_{1}, …, q_{d}}. It follows that τ(f^{d}) = 0. This completes the proof. □

Let {q_{1}, …, q_{d}} be given by the previous lemma, labelled so that −∞ ≤ q_{1} < q_{2} < ⋯ < q_{d} < ∞, and let l = 2d. We will begin by working with the group G^{l}. Note that every g ∈ G^{l} is orientation-preserving, has zero rotation number, and fixes every point in the set {q_{1}, …, q_{d}}. Throughout this section, we will be working on the intervals (q_{i}, q_{i+1}), where we adopt the convention that q_{d+1} = q_{1}.

Let M be a normal abelian subgroup of G^{l} which contains an element of infinite order. For the rest of the proof, fix a diffeomorphism g ∈ M which has infinite order.

Lemma 7.3. Let C(g) = {f ∈ G^{l} | gf = fg}. Then C(g) ≠ G^{l}.

Proof: A proof of this lemma is essentially contained in [FS]. This lemma is implied by the following theorem, which is classical.

Theorem 7.4. (Hölder’s Theorem) If a group of homomorphisms acts freely on ℝ, then it is abelian.

If f ∈ C(g), then Fix(f) = Fix(g). So on every interval (q_{i}, q_{i+1}), 1 ≤ i ≤ d, no element of C(g) has a fixed point. By Theorem 7.4, the restriction of the action of C(g) to each interval (q_{i}, q_{i+1}) is abelian. Since f(q_{i}) = q_{i} for all f ∈ C(g) and for all q_{i} ∈ Fix(g), C(g) is an abelian subset of G^{l}. But G^{l} is not abelian, so C(g) ≠ G^{l}. □

Lemma 7.5. Let f ∈ G^{l} \ C(g). Then for every interval (q_{i−1}, q_{i}] there is a positive real number λ_{i} and a C^{r} map α_{i} : (q_{i−1}, q_{i}] → ℝP^{1} with the following properties:

(1) α_{i}g(p) = α_{i}(p) + 1 and α_{i}f(p) = λ_{i}α_{i}(p).

(2) α_{i}|_{(q_{i−1},q_{i})} is a C^{r} diffeomorphism onto (−∞, ∞);

(3) There are an orientation-preserving C^{r} diffeomorphism h_{i} from a neighborhood of q_{i} to a neighborhood of ∞ and integers s_{i} ∈ {1, …, r − 1}, o_{i} ∈ {±1}, such that, for all p in this neighborhood:

α_{i}(p) = o_{i}h_{i}(p)^{s_{i}}

The same conclusions hold, with the same λ_{i}, o_{i} and α_{i}|_{(q_{i−1},q_{i})}, but different local (orientationreversing) diffeomorphism h_{i}^{∗} and integer s_{i}^{∗}, when q_{i} is replaced by q_{i−1} and the interval (q_{i−1}, q_{i}] is replaced by [q_{i−1}, q_{i}).

Remark: To ensure that the conditions λ_{i} > 0 (as opposed to λ_{i} ≠ 0) hold in Lemma 7.5, it is necessary that we chose l to be even.

Proof: We use the following fact, proved by Takens:

Theorem 7.6. ([Ta], Theorem 4) Let h : [0, 1) → [0, 1) be a C^{∞} diffeomorphism with unique fixed point 0 ∈ [0, 1). If h is not infinitely flat, then there exists a unique C^{∞} vector field X on [0, 1) such that h = h^{1}, where h^{t} is the flow generated by X.

For each 1 ≤ i ≤ d, Let g_{i}^{t} : (q_{i−1}, q_{i}] → (q_{i−1}, q_{i}] be the flow given by this theorem with g_{i}^{1} = g|_{(q_{i−1},q_{i}]}. If f ∈ G^{l} \ C(g), then since g ∈ M, we have fgf^{−1} ∈ M, and therefore fgf^{−1} ∈ C(g). By Lemma 3.7, for 1 ≤ i ≤ d, we must have

fgf^{−1} = g_{i}^{λ_{i}}

on (q_{i−1}, q_{i}], for some λ_{i} ∈ ℝ \ {0}, λ_{i} ≠ 1. Note that, because l is even, λ_{i} must be positive, for all i. So assumption D. of Section 3 holds in the interval (q_{i−1}, q_{i}] for each q_{i} ∈ {q_{1}, …, q_{d}}.

The same reasoning can be applied to [q_{i−1}, q_{i}), using a possibly different flow g̃_{i}^{t} and constant μ_{i} with

fgf^{−1} = g̃_{i}^{μ_{i}}.

Since g_{i}^{λ_{i}} and g̃_{i}^{μ_{i}} coincide on (q_{i}, q_{i+1}), it is not hard to see that we must have λ_{i} = μ_{i}. Now the result follows from Proposition 3.1, as in the proof of Lemma 5.6 and Corollary 5.8. □

Corollary 7.7. For every f ∈ G^{l} \ C(g), there is a positive real number λ = λ(f) ≠ 1 such that λ_{i} = λ, for all 1 ≤ i ≤ d, where λ_{i} is given by Lemma 7.5. For every i, s_{i} = s_{i+1}^{∗}, where addition is mod d.

Proof: Let 1 ≤ i ≤ d. As in the proof of Corollary 5.8, we have that g is conjugate in a left neighborhood of q_{i} to x ↦ x/(1 + x^{s_{i}})^{1/s_{i}} and g is conjugate in a right neighborhood of q_{i} to x ↦ x/(1 + x^{s_{i+1}^{∗}})^{1/s_{i+1}^{∗}}. Since g is C^{∞}, we must have s_{i} = s_{i+1}^{∗}. But then f is conjugate in a left neighborhood of q_{i} to x ↦ x/λ_{i}^{1/s_{i}} and f is conjugate in a right neighborhood of q_{i} to x ↦ x/(λ_{i+1})^{1/s_{i+1}^{∗}}. It follows that λ_{i} = λ_{i+1}. Set λ to be this common value. □

The proof of the next corollary is identical to the proof of Corollary 5.8.

Corollary 7.8. For every f ∈ G^{l} \ C(g), the map π : S^{1} → ℝP^{1} defined by:

π(p) = α_{i}(p), for p ∈ (q_{i−1}, q_{i}]

is a C^{r} ramified cover with signature s = (s_{1}, … s_{d}, o_{1}, …, o_{d}). The diffeomorphism f is a π-ramified lift of the map x ↦ λ(f)x, and g is a π-ramified lift of the map x ↦ x + 1.

Corollary 7.9. g embeds in a unique C^{r} flow g^{t}, with g = g^{1}. The elements of C(g) belong in the flow for g and, for each f ∈ G^{l} \ C(g), lie in the ramified lift under π_{f} of the translation group {x ↦ x + β | β ∈ ℝ}. That is, for any h ∈ C(g), there exist real numbers β, t such that h = g^{t} is a π_{f}-ramified lift of the map x ↦ x + β.

Proof: This corollary follows directly from Corollary 7.8, Lemma 5.9 and Lemma 5.10. □

Lemma 7.10. For any f_{1}, f_{2} ∈ G^{l} \ C(g), there exists a real number γ such that f_{2} is a π_{f_{1}}-ramified lift of the map x ↦ λ(f_{2})x + γ.

Proof: The proof is expressed in a series of commutative diagrams.

Lemma 7.11. There exists an α ∈ ℝ such that g^{α} is the (π_{f_{1}}, π_{f_{2}}, id)-ramified lift of the identity map.

Proof: The following diagram shows that if îd is the (π_{f_{1}}, π_{f_{2}}, id)-ramified lift of the identity map on ℝP^{1}, then îd ∘ g = g ∘ îd:

[figure: S^{1}; S^{1}; ℝP^{1}; ℝP^{1}; S^{1}; S^{1}; ℝP^{1}; ℝP^{1}; g; x ↦ x + 1; π_{f_{1}}; π_{f_{1}}; g; x ↦ x + 1; π_{f_{2}}; π_{f_{2}}; îd; îd; id; id]

Since g embeds in a flow g^{t} that is a ramified lift of an affine flow, it follows from Corollary 7.9 that there exists an α such that îd = g^{α}. □

Lemma 7.12. For every t_{0} ∈ ℝ there exists γ ∈ ℝ such that g^{t_{0}} is the (π_{f_{1}}, π_{f_{2}}, id)-ramified lift of the map x ↦ x − γ.

Proof: Let α be given by the previous lemma. Let γ be the real number such that g^{t_{0}−α} is a π_{f_{1}} - ramified lift of x ↦ x − γ. The proof follows from the following diagram:

[figure: S^{1}; S^{1}; ℝP^{1}; ℝP^{1}; S^{1}; ℝP^{1}; g^{t_{0}−α}; x ↦ x − γ; π_{f_{1}}; π_{f_{1}}; îd = g^{α}; id; π_{f_{2}}]

The composition of the maps on the top row is g^{α} ∘ g^{t_{0}−α} = g^{t_{0}}. The composition of the maps on the bottom row is x ↦ x − γ. □

Lemma 7.13. For all t ∈ ℝ, f_{2}g^{t}f_{2}^{−1} = g^{λ(f_{2})t}.

Proof: The proof follows from the following diagram:

[figure: S^{1}; S^{1}; ℝP^{1}; ℝP^{1}; S^{1}; ℝP^{1}; S^{1}; ℝP^{1}; f_{2}^{−1}; x ↦ \frac{1}{λ(f_{2})}x; π_{f_{2}}; π_{f_{2}}; π_{f_{2}}; g^{t}; x ↦ x + t; π_{f_{2}}; f_{2}; x ↦ λ(f_{2})x]

The composition of the maps on the bottom row gives x ↦ x + λ(f_{2})t. By uniqueness (Lemma 5.9), f_{2}g^{t}f_{2}^{−1} = g^{λ(f_{2})t} for all t ∈ ℝ. □

Let α be given by Lemma 7.11, and let γ be given by Lemma 7.12, with t_{0} = λ(f_{2})α. From the following diagram:

[figure: S^{1}; S^{1}; ℝP^{1}; ℝP^{1}; S^{1}; S^{1}; ℝP^{1}; ℝP^{1}; îd = g^{α}; id; π_{f_{1}}; π_{f_{2}}; g^{λα}; x ↦ x − γ; π_{f_{1}}; π_{f_{2}}; f_{2}; f_{2}; F; x ↦ λ(f_{2})x]

it follows that F (x) = λ(f_{2})x + γ, completing the proof of Lemma 7.10. □

Proposition 7.14. Fix f ∈ G^{l} \ C(g). Then for each h ∈ G, there exists F ∈ Aff(ℝ) such that h is a π_{f}-ramified lift of F.

Proof: By Corollary 7.9 and Lemma 7.10, we have that for each h ∈ G, there exists k ∈ Aff_{+}(ℝ), so that h^{l} is a π_{f}-ramified lift of k^{l}. Therefore, by Proposition 5.12, h is a π_{f}-ramified lift of either k or −k. □

By Lemma 4.5, this completes the proof of Theorem 1.9. □

Acknowledgments

Many useful conversations with Gautam Bharali, Christian Bonatti, Keith Burns, Matthew Emerton, Benson Farb, Giovanni Forni, John Franks, Ralf Spatzier, Jared Wunsch and Eric Zaslow are gratefully acknowledged. We thank Andrés Navas and Etienne Ghys for very useful comments on earlier versions of this paper, and for pointing out several references to us. Ghys supplied the simple proof of Proposition 2.1, and Curt McMullen supplied the simple proof of Lemma 4.1. Finally, we thank Benson Farb for reminding us that the BS groups are often interesting. The second author was supported by an NSF grant.

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Lizzie Burslem (burslem@umich.edu) Department of Mathematics, University of Michigan 2074 East Hall, Ann Arbor, MI 48109-1109 USA

Amie Wilkinson (wilkinso@math.northwestern.edu) Department of Mathematics, Northwestern University 2033 Sheridan Road, Evanston, IL 60208-2730 USA