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# Paper surfaces and dynamical limits

Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved June 18, 2010 (received for review February 19, 2010)

## Abstract

It is very common in mathematics to construct surfaces by identifying the sides of a polygon together in pairs: For example, identifying opposite sides of a square yields a torus. In this article the construction is considered in the case where infinitely many pairs of segments around the boundary of the polygon are identified. The topological, metric, and complex structures of the resulting surfaces are discussed: In particular, a condition is given under which the surface has a global complex structure (i.e., is a Riemann surface). In this case, a modulus of continuity for a uniformizing map is given. The motivation for considering this construction comes from dynamical systems theory: If the modulus of continuity is uniform across a family of such constructions, each with an iteration defined on it, then it is possible to take limits in the family and hence to complete it. Such an application is briefly discussed.

The most common way of constructing surfaces is by identifying sides of a plane polygon *P*. In this article the usual finiteness assumption is dropped, and surfaces are constructed by identifying infinitely many segments along the boundary of *P*. Conditions are presented under which the resulting space *S* is a 2-sphere; the metric on the sphere coming from the metric on *P* is described, and this is then used to investigate whether or not the complex structure coming from the Euclidean structure in the interior of *P* extends across the *scar* (the quotient of the boundary of *P*) to give *S* a Riemann sphere structure. A sufficient condition for the complex structure to extend uniquely to all of *S* is presented, and, when this condition holds, a modulus of continuity for the quotient map is given. This modulus of continuity makes it possible to show that a certain family of constructions of pseudo-Anosov maps (in the *unimodal generalized pseudo-Anosov* family) is equicontinuous and thus to find limits of sequences of sphere pseudo-Anosovs and a completion of this family. This addresses a question that D. Sullivan asked the first author and goes toward constructing the closure of the set of pseudo-Anosov maps relative to finite collections of points in a given surface.

This article contains a brief description of these results. Full details, and the extension to surfaces other than the sphere, will appear elsewhere.

## Paper-Folding schemes

Paper-folding schemes are the identification schemes around the boundaries of Euclidean polygons whose quotients are the objects of interest in this article. The metric structure of the identification spaces is of central importance, and for this reason metric quotients are used rather than the more familiar topological quotients.

This section starts with a brief discussion of quotient metrics and intrinsic metrics. See ref. 1 for more details. The reader seeking a concrete example of the type of quotient metric space that will be considered can look ahead to Example 6.

(Quotient metrics). Let (*X*,*d*) be a metric space and R be a reflexive and symmetric relation on *X* (or think of R as a collection of subsets of *X* whose associated relation is *x*R*y* iff *x*, *y* lie in the same set in the collection R). An R-*chain* from *x* to *y* is a sequence in *X*^{2} such that *x*R*p*_{0}, *q*_{i}R*p*_{i+1} for *i* = 0,…,*k* - 1, and *q*_{k}R*y*. Its *length* is . (This means that to get from *x* to *y* with an R-chain one pays to move *between* sets in R but not *within* them.)

Define by *d*^{R}(*x*,*y*)≔ inf{L^{R}((*p*_{i},*q*_{i})): ((*p*_{i},*q*_{i})) is an R-chain from *x* to *y*}. Then *d*^{R} is a semimetric on *X*. The equivalence relation that identifies points at *d*^{R}-semi-distance 0 is denoted and the quotient space under this equivalence relation is the *quotient metric space* of (*X*,*d*) under the relation R, denoted (*X*/*d*^{R},*d*^{R}).

(Lengths and intrinsic metrics). Let (*X*,*d*) be a metric space. The *length* of a path *γ*: [*a*,*b*] → *X* is , where the supremum is taken over all finite partitions *a* = *t*_{0} < *t*_{1} < ⋯ < *t*_{k} = *b* of the interval [*a*,*b*].

A metric is *intrinsic* if the distance between two points is arbitrarily well approximated by lengths of curves joining the two points. The metric is *strictly intrinsic* if the distance is attained: That is, if for every *x*,*y*∈*X*, there exists a path from *x* to *y* whose length equals *d*(*x*,*y*) (in particular, *d*(*x*,*y*) = ∞ if there is no path from *x* to *y*). It can be shown that a compact intrinsic metric is strictly intrinsic (see ref. 1).

If *d* is not an intrinsic metric, then there is an *induced intrinsic metric* on *X*, defined by . For example, if then there is an intrinsic metric *d*_{P} on *P* induced by the Euclidean metric on , which need not of course agree with the subspace metric on *P*.

(Segment pairing). Let be a *multipolygon* (a union of finitely many mutually disjoint polygons) and *C* = ∂*P* be its boundary, with each component of *C* oriented positively. Let *α*, *α*^{′}⊂*C* be closed nontrivial segments of the same length with disjoint interiors. The *segment pairing* 〈*α*,*α*^{′}〉 is the equivalence relation that identifies pairs of points of *α* and *α*^{′} in a length-preserving and orientation-reversing way. The segments *α*, *α*^{′}, and any two points that are identified under the pairing are said to be *paired*. Two paired points that lie in the interior of a segment pairing form an *interior pair*. Note that 〈*α*,*α*^{′}〉 and 〈*α*^{′},*α*〉 represent the same pairing.

Two segment pairings 〈*α*,*α*^{′}〉, 〈*β*,*β*^{′}〉 are *interior disjoint* if the interiors of all four segments are disjoint.

The *length* of a segment pairing 〈*α*,*α*^{′}〉, denoted |〈*α*,*α*^{′}〉|, is the length of one of the segments in the pairing, i.e., . If is a (countable) collection of pairwise interior disjoint segment pairings, its *length*, denoted , is the sum of the lengths of the pairings in , i.e., .

A collection of interior disjoint pairings is *full* if equals half the length of *C*. This means that the pairings in cover all of *C* up to a set of Lebesgue 1-dimensional measure zero.

A pairing of two segments that have one endpoint in common is a *fold*, and the common endpoint is its *folding point*. Thus the folding point in a fold is alone in its equivalence class.

(Paper-folding scheme). A *paper-folding scheme*, or simply a *paper folding*, is a pair where is a multipolygon with the intrinsic metric *d*_{P} induced from , and is a full collection of interior disjoint segment pairings on ∂*P*. The metric quotient of *P* under the semimetric induced by the pairing relation is the associated *paper space*. If it is known that *S* is a closed (compact without boundary) topological surface, then is a *surface paper-folding scheme* and *S* is the associated *paper surface*.

The projection map is denoted *π*: *P* → *S*, and the quotient *G* = *π*(∂*P*)⊂*S* of the boundary is the *scar*. Notice that the restriction *π*: Int(*P*) → *S* ∖ *G* is a homeomorphism.

For , a point *x*∈*G* is a *vertex of valence* *k* or simply a *k*-*vertex* if either #*π*^{-1}(*x*) = *k* ≠ 2, or #*π*^{-1}(*x*) = *k* = 2 and *π*^{-1}(*x*) contains a vertex of *P*. Points of *G* that are neither vertices nor accumulations of vertices are *planar points*. A *k*-vertex, with *k* ≠ ∞, which is not an accumulation of vertices is a *regular* vertex. A point is *singular* if it is an ∞-vertex or is an accumulation of vertices. Let be the set of all vertices, denote the set of regular vertices and be the set of singular points (vertices or not) of *G*, so that . The closure of a connected component of is an *edge* of *G*.

Points in *π*(Int(*P*)) are also called *planar*. Thus the set of all planar points in *S* is the complement of .

The focus of this article is on paper-folding schemes whose quotient is a surface. For clarity of exposition, attention will be concentrated on *plain* foldings. These are both the most common and the simplest type of paper foldings, and for them the paper space is always a sphere and the scar is always a dendrite (Theorem 7).

(Plain folding). A paper-folding scheme is *plain* if *P* is a single polygon, and there are no four distinct points *x*, *y*, *x*^{′}, *y*^{′} on ∂*P*, in the given order around ∂*P*, with and , i.e., pairs of paired points are unlinked along ∂*P*.

Consider the unit square and pick a decreasing sequence of real numbers such that . Define the following segment pairings on ∂*P* (Figs. 1 and 2). The two vertical sides are paired and the top side is folded in half. On the bottom side define countably many folds of lengths *a*_{i}, pairwise disjoint except possibly at endpoints.

One way of doing this (Fig. 1) is by placing the folds with decreasing lengths from right to left, with the left endpoint of each coinciding with the right endpoint of the next. Then the complement of the bottom pairings is the bottom left vertex of *P*. In this case the equivalence relation induced by , besides identifying paired points, identifies together all fold endpoints on the bottom side with the bottom left vertex of *P*.

Another way of arranging the folds (Fig. 2) is by taking them to be pairwise disjoint. In this case, the closure of the complement is a Cantor set *K* of Lebesgue measure 0, and the metric quotient identifies all points of *K*. Clearly, no matter the way chosen to place the folds along the bottom side of *P*, these are plain paper-folding schemes.

(Plain paper-folding structure theorem). The quotient *S* of a plain paper-folding scheme is a topological sphere, and its scar *G* is a dendrite. Moreover, the set of planar points is open and dense in *G*, while the set of vertices and singular points is a closed nowhere dense subset of *G* with zero measure (under the push-forward of Lebesgue 1-dimensional measure on ∂*P*).

The main tool used in the proof is Moore’s theorem on quotients of the 2-sphere (2, 3). The plainness condition ensures that *S* has genus 0.

The metric structure of paper spaces is now considered.

(Cone). Let *X* be a topological space. The *cone* Cone(*X*) over *X* is the (topological) quotient of [0,∞) × *X* by the equivalence relation which collapses {0} × *X* to a point, that is, whose only nontrivial class is {0} × *X*. This point in the quotient is the *origin* or *apex* of the cone. If (*X*,*d*) is a metric space, then it is possible to make Cone(*X*) into a metric space by defining a distance as follows: if *p* = [*t*,*x*], *q* = [*s*,*y*]∈Cone(*X*), set

(Conic-flat surface). A *conic-flat surface* *S* is a metric space that is locally isometric to cones on circles, i.e., for every *p*∈*S* there exist *r*, *ε* > 0, , and an isometry from *B*_{S}(*p*,*ε*) onto , where is a circle in of radius *r* with the intrinsic metric. This means that there are two types of points on a conic-flat surface *S*: those at which *S* is locally Euclidean and those where *S* is locally isometric to a neighborhood of the apex of . In the latter case, 2*πr* is the *cone angle* at the point.

The metric structure of a paper space is summarized in the following theorem. The proof, while technical, uses standard methods.

Let be a paper-folding scheme and *S* be the associated paper space. Then the quotient metric on *S* is strictly intrinsic and is a conic-flat surface.

## The Conformal Structure on Paper Surfaces

The Euclidean structure induces a complex structure in the neighborhood of all planar points. This complex structure extends uniquely across *k*-vertices with *k* ≠ ∞. In this section, the question of when the complex structure extends across an isolated singularity is addressed. That is, when there is a complex structure on a neighborhood of the singularity that is compatible with that on . Using a theorem of McMullen (4), it is possible to apply similar techniques to nonisolated singularities.

(Planar radius, *n*(*q*; *r*)). Let *G* be the scar of a paper-folding scheme. Given *q*∈*G* and *r*≥0 let *C*_{G}(*q*; *r*) be the set of points of *G* at distance exactly *r* from *q* (in the intrinsic metric on *G* as a subset of *S*). A radius *r* > 0 is *planar for* *q* if all points of *C*_{G}(*q*; *r*) are planar.

The number *n*(*q*; *r*) is defined by .

If *r* > 0 is a planar radius for *q*∈*G* then *n*(*q*; *r*) is finite and is a locally constant function of both variables *q*, *r*. Moreover, given *q*∈*G*, the set of radii which are not planar for *q* has zero measure.

The following theorem gives conditions under which the complex structure extends uniquely across an isolated singularity:

Let be a surface paper-folding scheme with associated paper surface *S* and scar *G*⊂*S*. If *q*∈*G* is an isolated singular point then the complex structure on extends uniquely across *q* provided that the following condition holds: [1]where m_{G} is the push-forward to *G* of Lebesgue measure on ∂*P* and *B*_{G}(*q*; *r*) is the metric ball in *G* of radius *r* centered at *q*. In particular, if all singular points are isolated and the integral condition above holds for each of them, then *S* has a closed Riemann surface structure.

Observe that the integral in the statement diverges at planar points in *G* and at regular vertices: At a *k*-vertex, with *k* < ∞, or at a planar point, for *r* small enough, *n*(*q*; *r*) is constant equal to *k* and m_{G}(*B*_{G}(*q*; *r*)) = 2*kr* (*k* = 2 at planar points) so that the integral is comparable to for *r*↓0.

A brief sketch of the proof, and a survey of the techniques from geometric function theory that it uses, will now be provided.

### Modulus of an Annulus.

A *topological annulus* is a domain whose complement has exactly two connected components. It is a theorem in the theory of conformal mapping that every topological annulus can be mapped conformally onto a round annulus and that this mapping is unique up to postcomposition with a homothety. It follows that, if 0 < *r*_{1} < *r*_{2} < ∞, the ratio *r*_{2}/*r*_{1} is invariant under conformal mapping. The *modulus* mod(Ω) of the domain is defined by So-called “length-area methods” allow one to prove subadditivity of the modulus: If Ω is such a topological annulus, and (Ω_{j}) is a finite or countably infinite sequence of mutually disjoint topological annuli *nested* in Ω (i.e., the inclusions Ω_{j} → Ω are homotopy equivalences), then .

Now let *S* be a closed surface, and suppose that there is a set Λ⊂*S* such that *S*^{′} = *S* ∖ Λ has a Riemann surface structure, i.e., it is covered by coordinate charts whose transition functions are holomorphic. Let *q*∈Λ be an isolated point of Λ and *U*⊂*S* be an open disk containing *q* but no other points of Λ. Then *U*^{′} = *U* ∖ {*q*} is a *planar* Riemann surface (one on which every simple closed curve separates), and it is a theorem from Riemann surface theory that it is conformally homeomorphic to a subset of the plane. Because *U*^{′} is a topological annulus it has a modulus and, choosing *U* to be smaller if necessary, it is conformally isomorphic to an annulus *A*(*r*_{1},*r*_{2}) with *r*_{2} < ∞.

If it can be shown that mod(*U*^{′}) = ∞ then it follows that *r*_{1} = 0, and a conformal isomorphism *φ*^{′}: *U*^{′} → *A*(0,*r*_{2}) extends uniquely to a homeomorphism by setting *φ*(*q*) = 0. Moreover, Riemann’s removability theorem says that if the construction is repeated with a different disk *V*, yielding a homeomorphism , then *ψ*∘*φ*^{-1} is conformal at 0 as well. That is, it is possible to extend the conformal structure on *S*^{′} *uniquely* across *q*.

The crucial point in this argument is to be able to show that mod(*U*^{′}) = ∞. To do this, one can find a sequence (Ω_{j}) of nested subannuli of *U*^{′} with divergent modulus sum. This is precisely what is ensured by the divergent integral [**1**] in the statement of Theorem 12.

To understand the relationship between the divergent sum used in the argument above and the divergent integral [**1**], another result using length-area methods is required. Let be a topological annulus with boundary components *C*_{1} and *C*_{2}, and let Γ be the set of paths γ in Ω that start on *C*_{1} and end on *C*_{2}. Given any conformal metric *ρ* = *λ*(*z*)|*dz*|, where *λ*(*z*) is a nonnegative Borel measureable function, define where *ℓ*_{ρ}(*γ*) = ∫_{γ}*λ*(*z*)|*dz*|, and define Then where *ρ* ranges over all conformal metrics on Ω. An immediate consequence is that for any given conformal metric *ρ*.

Now let *C*^{1},*C*^{2},…,*C*^{n} be simple closed curves in Ω such that *C*^{1} separates *C*_{1} and *C*_{2}, and *C*^{j+1} separates *C*^{j} and *C*_{2} for each *j*; and let Ω_{j} be the topological annulus bounded by *C*^{j} and *C*^{j+1}. If Ω_{j} is “*ρ*-thin”, then Area_{ρ}(Ω_{j}) ≈ *d*_{ρ}(*C*^{j},*C*^{j+1})*ℓ*_{ρ}(*C*^{j}), so that Passing to the limit as the family {*C*^{j}} gets denser and denser in Ω, In the case of a paper surface, the lengths of the curves *C*^{r} can be controlled by the lengths of pieces of the scar: A family of simple closed curves *C*^{r} is constructed zooming into the singularity *q*, whose lengths are comparable to m_{G}(*B*_{G}(*q*; *r*)) + *r*·*n*(*q*; *r*), from which the criterion [**1**] can be derived.

### ∞-od singularities.

A case of particular interest is that of ∞-od singularities: They have already appeared in Example 6 and reappear in dynamical applications.

(∞-od singularity). Let be a sequence of positive real numbers converging to 0 and consider the intervals [0,*a*_{n}], *n* = 1,2,…. The metric quotient of these intervals under the equivalence relation that identifies together the points 0 in all of them is an ∞-*od*. The point in the quotient corresponding to the nontrivial equivalence class is the *apex* of the ∞-od. An isolated ∞-vertex *q*∈*G* is an ∞-*od singularity* if there exists *r*_{0} > 0 such that *B*_{G}(*q*; *r*_{0}) is an ∞-od with apex *q*.

Suppose that *q*∈*G* is an ∞-od singularity. It follows that *B*_{G}(*q*; *r*) is an ∞-od for every 0 < *r* ≤ *r*_{0}. Thus every point *x*∈*C*_{G}(*q*; *r*) is joined to *q* by a unique arc all of whose interior points are planar and whose length is *r*. This implies that . There are as many such arcs in *B*_{G}(*q*; *r*) as there are points in *C*_{G}(*q*; *r*) from which it follows that m_{G}(*B*_{G}(*q*; *r*))≥2*r*·*n*(*q*; *r*). This establishes the following

For the complex structure on to extend uniquely across an ∞-od singular point *q*∈*G* it is sufficient that [2]

Consider again Example 6. Notice that in the hypotheses of Corollary 15 only the measure m_{G}(*B*_{G}(*q*; *r*)) is taken into account, and the way in which the identifications about *q* are carried out is not relevant. Thus there is no distinction between the two ways of arranging the folds in the example.

Defining *N*(*r*)≔ max{*n*: *a*_{n}≥*r*}, [3](the factor 2 appears because measure and distance are related by this factor). Whether or not condition [**2**] holds depends on the behavior of the sequence (*a*_{n}).

Suppose, for example, that *a*_{n}*≍*1/*n*^{k}, for some *k* > 1: that is, *C*_{1}/*n*^{k} ≤ *a*_{n} ≤ *C*_{2}/*n*^{k} for some positive constants *C*_{1} and *C*_{2}. In this case it can be shown that m_{G}(*B*_{G}(*q*; *r*))≥*C*_{3}*r*^{1-1/k} for sufficiently small *r*, so that the integral [**2**] converges. This means the criterion cannot be used to guarantee that the complex structure extends uniquely across the point *q* (the authors do not know whether or not it extends uniquely).

If, on the other hand, *a*_{n}*≍*1/*λ*^{n} for some *λ* > 1, then [**3**] gives for *r* sufficiently small, so that the integral [**2**] diverges. Therefore Corollary 15 applies and the quotient space is a complex sphere.

If the Cantor construction of Example 6 is that of the standard middle thirds Cantor set, then listing the edges of the ∞-od in decreasing order gives and hence Thus this is an example in which the hypothesis of Corollary 15 is not satisfied.

In fact it is possible to construct examples in which there is a Cantor set of singularities with the property that the complex structure extends uniquely across all of them.

## Global Modulus of Continuity

In this section attention is again restricted to plain paper foldings and paper spheres, and it is shown how to obtain a global modulus of continuity for a uniformizing map. It will be assumed therefore that is a plain paper folding and that there are finitely many singular points in the scar *G*, at each of which the hypothesis of Theorem 12 is satisfied. It thus follows from that theorem that the complex structure extends uniquely across all singularities, so *S* is a sphere and has a complex structure: By the Uniformization Theorem *S* is conformally isomorphic to the Riemann sphere . To fix one such isomorphism, pick points and , where *P*_{h} is what remains of *P* after taking away a collar neighborhood of width *h* of ∂*P*, for some *h* > 0 that is small enough that *P*_{h} is a connected open set. Set and and define to be the isomorphism such that *u*(*p*_{0}) = 0, *u*(*p*_{∞}) = ∞ and the composition , *ϕ*≔*u*∘*π* satisfies *ϕ*^{′}(*p*_{∞}) = 1.

The uniformizing map , *ϕ*≔*u*∘*π*, has a modulus of continuity which depends only on the scar and its metric, and on geometric constants of the polygon.

By *geometric constants* of the polygon *P* is meant: (*i*) the supremum of *h* for which *P* has an inner collar of width *h* that is also a polygon with as many sides as *P*; and (*ii*) how close the interior angles of *P* get to 0 or 2*π*.

At points *q* in the scar *G* the modulus of continuity is given by [4]where and the constants *C*_{1}, *C*_{2}, *b*, and depend on the geometry of *P* and on the metric on *G*. At other points, the formula for *ρ* is more complicated and depends on more technical details but is of the same general form.

The proof uses the Grötzsch Modulus Theorem to obtain estimates on the distance between two points from the modulus of an annulus separating them from a given circle (5, 6).

## Dynamical Limits

Thurston’s classification of surface homeomorphisms (7) states that every isotopy class of surface homeomorphisms contains a representative of one of three types: *finite order* (some iterate is the identity); *reducible* (there is an invariant collection of mutually disjoint essential simple closed curves); or *pseudo-Anosov* (there is a transverse pair of measured invariant singular foliations, one of which is expanded by a factor *λ* > 1 and the other of which is contracted by a factor 1/*λ*). There are two main properties of pseudo-Anosov homeomorphisms that are important in this article. First, they have minimal dynamics in their isotopy class. This minimality can be expressed in several different ways, but perhaps the most compelling is Handel’s result (8) that if *ψ*: *S* → *S* is pseudo-Anosov and *f* is isotopic to *ψ*, then there is a closed *f*-invariant subset *X* of *S* and a continuous surjection *π*: *X* → *S* such that *ψ*∘*π* = *π*∘*f*|_{X}: in particular, pseudo-Anosovs minimize topological entropy in their isotopy class. Second, pseudo-Anosovs admit Markov partitions, so that although the dynamics of a pseudo-Anosov is complicated, there are tools for understanding it.

In fact, the Markov partition of a pseudo-Anosov provides a method to construct its surface of definition as a finite paper surface—see Fig. 3, which illustrates this for a pseudo-Anosov homeomorphism *ψ* of the five-punctured sphere. The multipolygon is a union of four rectangles (Markov boxes); the identification scheme is represented by means of dotted lines on the horizontal edges and digits on identified segments of vertical edges; and *ψ* stretches each box uniformly by a factor *λ* > 1 in the vertical direction, and contracts it uniformly by a factor 1/*λ* in the horizontal direction. *ψ* therefore has an invariant *stable* foliation of horizontal leaves that it contracts and an invariant *unstable* foliation of vertical leaves that it expands. These foliations have five one-pronged singularities (at the punctures, which are marked with solid circles), and one three-pronged singularity (at the three identified points marked with a solid square). Notice that, while there are only finitely many segment pairings, the number of pairings increases with the number of punctures by Poincaré-Hopf index considerations.

In this section, brief details are given of an application of the techniques presented in this paper to the family of *generalized pseudo-Anosovs* associated with tent maps of the interval. These maps are defined like pseudo-Anosovs, but their foliations can have infinitely many singularities so that their surfaces of definition are given as paper surfaces with infinitely many segment pairings. The article then ends with a discussion of the type of application which motivated this work.

### Limits of Unimodal Generalized Pseudo-Anosovs.

Consider the family of *tent maps* *T*_{λ}: [0,1] → [0,1], where , defined by Let be the (countable dense) subset of parameters *λ* for which the critical point 1 - 1/*λ* is either periodic or preperiodic. In ref. 9, a *generalized pseudo-Anosov map* *ψ*_{λ}: *S*_{λ} → *S*_{λ} is constructed for each *λ*∈Λ: It is a quasi-conformal homeomorphism of a complex sphere *S*_{λ} whose dynamics mimics that of *T*_{λ} (apart from a few well-understood exceptions, *ψ*_{λ} is a Teichmüller mapping associated to a quadratic differential with infinitely many zeros and poles). The spheres *S*_{λ} are expressed as paper surfaces—a typical example is shown in Fig. 4. The multipolygon is a single polygon *P*_{λ}, shown at the top of the figure. There are infinitely many segment identifications on its boundary: The top and bottom edges of each of the two “tongues” are identified, and around the remainder of the boundary the pattern of identifications shown at the bottom of the figure is repeated multiple times (in each occurence of the pattern, the lengths of the identified segments decrease exponentially, which means in particular that the integral condition [**1**] holds at each singularity). The action of *ψ*_{λ} is to stretch *P*_{λ} uniformly by a factor *λ* in the horizontal direction and contract it uniformly by 1/*λ* in the vertical direction: The left side of *P*_{λ} (shaded light) is sent to the bottom of *P*_{λ}, while the right side (shaded dark) is rotated by *π* and sent to the top of *P*_{λ}. *ψ*_{λ} therefore preserves a horizontal unstable foliation and a vertical stable foliation, but these foliations have infinitely many singularities. The action in the horizontal direction is exactly that of the corresponding tent map *T*_{λ}. (This example corresponds to the periodic kneading invariant (100101*C*)^{∞}, with *λ* ≃ 1.686.)

One would like to be able to complete this family by defining *ψ*_{λ}: *S*_{λ} → *S*_{λ} for all , and the natural way of doing this is by taking limits through sequences *ψ*_{λi} with *λ*_{i}∈Λ. Theorem 18 provides a modulus of continuity for the uniformizing map and, provided that a uniform modulus of continuity can be found for all *λ*_{i}, Arzelà-Ascoli type arguments can be used to guarantee the existence of the limit. (One interesting example, the simplest in this context, is that of the “tight horseshoe,” which is obtained in the limit *λ* → 2. It is a version of Smale’s horseshoe map in which all of the gaps in the nonwandering Cantor set have been collapsed, leaving a homeomorphism whose nonwandering set is the whole sphere.)

This approach does not work for all convergent sequences *λ*_{i} because it is possible for the tongues in the polygons *P*_{λi} to become more and more dense, meaning that the polygons cannot be uniformly collared. However, for a certain class of sequences *λ*_{i} the polygons have no tongues, and by means of these sequences it is possible to construct uncountably many limiting Teichmüller mappings of the sphere corresponding to values *λ*∉Λ.

### Aspirational Example: The Hénon family.

Thurston’s classification theorem is often used to obtain lower bounds on the dynamics of specific homeomorphisms, such as those of the *Hénon family* *f*(*x*,*y*)≔(*a* - *x*^{2} - *by*,*x*), where *a*, *b* are real parameters (10). Let *P* be a finite union of periodic orbits of a specific Hénon map *f*. If the isotopy class of *f* in the plane punctured at the points of *P* is pseudo-Anosov, then the corresponding pseudo-Anosov homeomorphism *ψ*_{P} has minimal dynamics in the isotopy class: In particular, it provides a lower bound for the dynamics of *f*. A natural approach to understanding the dynamics of *f* is to take an increasing sequence (*P*_{n}) of such unions of periodic orbits, exhausting all of the periodic orbits of *f*, and take the limit of the sequence *ψ*_{Pn} of associated pseudo-Anosovs.

An immediate problem is how such a limit can be taken, given that the pseudo-Anosovs *ψ*_{Pn} are all defined on different abstract spheres . Theorem 18 provides a solution to this problem: Once a choice of normalization has been made, the spheres are all identified with the Riemann sphere by uniformizing maps . Provided that the geometry of the multipolygons is bounded, these uniformizing maps have a uniform modulus of continuity, so that the limiting sphere can be constructed.

Providing a method for taking such limits was the original motivation for the work described in this article, although understanding the Hénon family in this way is still not feasible, not least because the periodic orbit structure of Hénon maps is not sufficiently well-understood.

## Acknowledgments

The authors are very grateful to Fred Gardiner, who made several useful comments on an early draft of this article, and to the referee, whose detailed comments have much improved the exposition. The authors gratefully acknowledge the support of Fundação de Amparo à Pesquisa do Estado de São Paulo Grant 2006/03829-2. The first author would also like to acknowledge the support of Conselho Nacional de Desenvolvimento Científico e Tecnológico Grant 151449/2008-2 and the hospitality of Instituto Nacional de Matemática Pura e Aplicada, where part of this work was developed.

## Footnotes

^{1}To whom correspondence should be addressed. E-mail: T.Hall{at}liv.ac.uk.Author contributions: A.d.C. and T.H. performed research; and A.d.C. and T.H. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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