aa r X i v : . [ m a t h . D S ] F e b CUTTING SEQUENCES ON SQUARE-TILED SURFACES
CHARLES C. JOHNSON
Abstract.
We characterize cutting sequences of infinite geodesics on square-tiled surfaces by considering interval exchanges on specially chosen intervals onthe surface. These interval exchanges can be thought of as skew products overa rotation, and we convert cutting sequences to symbolic trajectories of theseinterval exchanges to show that special types of combinatorial lifts of Sturmiansequences completely describe all cutting sequences on a square-tiled surface.Our results extend the list of families of surfaces where cutting sequences areunderstood to a dense subset of the moduli space of all translation surfaces. Introduction
Given a surface with a Riemannian metric and a labeled collection of curves onthe surface, the cutting sequence of a geodesic is a biinfinite sequence of labels givenin the order in which the geodesic intersects the corresponding curves. The mostobvious question to consider concerns characterizing precisely which sequences oflabels correspond to cutting sequences.This question has been studied for a few special types of translation surfaces,which are surfaces equipped with a flat metric with cone-type singularities andtrivial holonomy. In particular, cutting sequences have been described for flat tori[MH38] [Ser85]; the regular polygon surfaces with an even number of sides werestudied by Smillie and Ulcigrai, [SU11]; Davis later extended the methods of Smillieand Ulcigrai to double n -gon surfaces with odd n [Dav13]; the L -shaped surfacebuilt from three squares was studied by Wu and Zhong, [WZ15]; and most recentlythe family of Bouw-M¨oller surfaces was studied by Davis, Pasquinelli, and Ulcigrai[DPU15]. Most of the previous results are obtained by considering the action of anelement of the Veech group (group of affine symmetries) of the surface. In [Dav14],for example, the effect of flip-shears on cutting sequences on some special types ofsurfaces are considered. In this paper we add one more family of surfaces to thislist by characterizing cutting sequences on square-tiled surfaces.Our approach will be to convert the question of which sequences are cuttingsequences into a question about the languages determined by a special class ofinterval exchange transformations which can be described as a certain type of skewproduct over a rotation. Languages of the family of interval exchanges with a fixedpermutation were studied in [FZ08]. Date : September 15, 2018.1991
Mathematics Subject Classification.
Primary 37E35.
Key words and phrases.
Cutting sequences, dynamical systems, square-tiled surfaces, andtranslation surfaces.The author thanks the referee for their careful reading of the original manuscript and for theirmany helpful remarks for improving the exposition.
After recalling the pertinent background and establishing some vocabulary andnotation we will prove that all cutting sequences on a given square-tiled surface canbe described as sequences of pairs with the first entry in each pair giving the label ofsome square on the surface, and the second entry giving an edge of the square. Thesecond entries of these pairs must form a cutting sequence on a torus, and the firstentries must form a sequence of labels that is consistent with the way the varioussquares on the surface are glued together. The sequences of pairs satisfying theserequirements have a relatively simple combinatorial description, but not all suchsequences are cutting sequences on the surface as this collection includes sequenceswhich would describe a geodesic which hits a conical singularity on the surface.We will show there is a combinatorial description of these sequences by consideringwalks on a family of graphs associated to the surface.We describe the necessary background on square tiled surfaces, Sturmian se-quences, languages, and interval exchanges in Section 2. We then show in Section 3that interval exchanges for some specially chosen intervals on a square-tiled sur-face are simply skew products over rotations which greatly simplifies understandingthese interval exchanges. In Section 4 we introduce the notion of consistency be-tween sequences of edges and the gluings of the surface and describe symmetricSturmian sequences. The complete statement and proof of the characterizationtheorem appears in Section 5. 2.
Background
Translation surfaces and the Veech dichotomy. A translation surface is a closed, oriented surface equipped with a geometry which is flat away from adiscrete set of conical singularities where small neighborhoods of a singularity areisometric to a Euclidean cone of cone angle 2 πn with n ≥ X, ω ) where X is a compact Riemannsurface and ω is a holomorphic 1-form on X . The equivalence between these twodefinitions comes from endowing ( X, ω ) with a singular flat metric by integrating ω to obtain an atlas of charts where transitions are given by translations (hence thename translation surface ), and so the usual Euclidean metric of C can be pulledback to a metric on the surface. Choosing a geodesic triangulation of the surfacewhere all singularities occur as vertices of the triangulation then gives a descriptionof ( X, ω ) as a collection of polygons with edge gluings.The set of all translation surfaces where the 1-form ω has the orders of its zerosfixed is called a stratum and is equipped with the structure of a complex orbifoldby assigning local coordinates by integrating ω along the relative cycles of eachsurface’s first homology H ( X, { p , p , ..., p n } ; C ), relative with respect to the zerosof the 1-form. The group GL(2 , R ) acts on each stratum by deforming the polygonalrepresentation of a surface. Since these deformations are linear, parallel lines arepreserved and so the deformed polygon still glues together to give a translationsurface in the same stratum.The stabilizer of a surface ( X, ω ) under the GL(2 , R ) action is called the Veechgroup of the surface and is denoted SL(
X, ω ). It was shown in [Vee89] that whenSL(
X, ω ) is a lattice, then for every direction the geodesic flow on (
X, ω ) satisfies
UTTING SEQUENCES ON SQUARE-TILED SURFACES 3 the following dichotomy: the flow is either completely periodic or uniquely ergodic.It had been previously shown in [KMS86] that for every translation surface the flowin almost every direction is uniquely ergodic, but there are surfaces where the flowin some directions is minimal but not uniquely ergodic.
Veech surfaces , the surfaceswith a lattice stabilizer, are surfaces where this is impossible: every non-periodicdirection is uniquely ergodic.More information about translation surfaces can be found in the surveys [MT02],[Zor06], and [Wri15].2.2.
Square-tiled surfaces.
Let T denote the standard square torus, C / Z [ i ]equipped with the 1-form dz . Notice that the torus is a Veech surface with Veechgroup SL(2 , Z ). A square-tiled surface is a covering π : X → T branched over atmost one point. The cover X is then given the translation structure correspondingto ω = π ∗ ( dz ). We can describe X as a collection of unit squares, the number ofsquares being the degree of the covering, glued together so that the top of eachsquare is identified with the bottom of another (or the same) square, and simi-larly for left- and right-hand edges of the squares. The gluings of the squares aredetermined by the covering’s monodromy.Every square-tiled surface determines, and is determined by, a pair of permuta-tions. Supposing the surface is made of d squares, let Λ = { , , ..., d } be the labelsof the squares. We then consider two permutations, h and v , of Λ and consider thesurface obtained by gluing the right-hand edge of square λ to the left-hand edge ofsquare h ( λ ), and the top of square λ to the bottom of square v ( λ ). This surfacewill be connected if and only if the group h h, v i acts transitively on Λ, and we willonly consider connected surfaces. 12 3 45 6 Figure 1.
The square-tiled surface on six squares with permuta-tions h = (1)(2 3 4)(5 6) and v = (1 2)(3 5)(4 6).In [GJ00] it was shown that when one translation surface is a cover of another, π : ( X, ω ) → ( Y, η ), their Veech groups must be commensurable. (In fact, GabrielaSchmith¨usen has developed an algorithm for determining the Veech group of asquare-tiled surface [Sch04].) Thus (
X, ω ) is a Veech surface if and only if (
Y, η )is a Veech surface. Hence square-tiled surfaces are Veech surfaces since they arecovers of tori.2.3.
Sturmian sequences.
The flat torus ( T , dz ) can be thought of as a singleunit square with its top and bottom edges identified, and its left- and right-handedges identified as well. If we label the horizontal edge V and the vertical edge H ,then a cutting sequence on the torus is a biinfinite sequence of H ’s and V ’s. CHARLES C. JOHNSON
Remark.
We point out that our convention of having V ’s on the top and bottomof the square and H ’s on the left- and right-hand sides may seem counter-intuitive,but does offer two small advantages over the more intuitive choice of putting V ’son the vertical edges and H ’s on the horizontal edges.We will use this same convention to describe labeled edges on square-tiled sur-faces, and in that situation crossing an H -edge corresponds to applying the permu-tation h ; and crossing a V -edge corresponds to applying the permutation v , i.e. aside labeled by H is shared between two squares glued by a horizontal translation.Additionally, with this convention, seeing more H ’s (respectively, V ’s) in a cut-ting sequence means the trajectory is more horizontal (resp. vertical) than vertical(horizontal).Not all sequences of H ’s and V ’s can be obtained as cutting sequences. Forexample, it is easy to see that the sequence will have isolated V ’s separated byseveral H ’s if the slope of the geodesic is less than 1 as in Figure 2. Having isolated V ’s (or H ’s) is not enough to be a cutting sequence, however. If the slope m is lessthan 1, for example, then the number of H ’s separating V ’s may be either (cid:4) m (cid:5) or (cid:4) m (cid:5) + 1 with the number of V ’s in each block depending on where the geodesic lastintersected the horizontal edge labeled V . VVH H
Figure 2.
The cutting sequence for the geodesic shown here is
HV HHV HHHV H... .As tori are Veech surfaces, all geodesics in a given direction will be periodic oruniformly distributed throughout the surface, each case being determined by theslope of the geodesic. If the slope is rational, then the geodesic is periodic. Ifthe slope is irrational, then the geodesic will be uniformly distributed. These twopossibilities are reflected in the types of cutting sequences which appear. Periodicgeodesics of course give periodic cutting sequences, while uniformly distributedgeodesics give
Sturmian sequences .A Sturmian sequence is a sequence on two characters that is not eventuallyperiodic and whose complexity function, which counts the number p ( n ) of distinctsubwords of a given length n , has the smallest possible value for a non-eventuallyperiodic sequence: p ( n ) = n + 1. Sturmian sequences are described in detail in[Arn02].We can determine if a given sequence in the alphabet E = { H, V } is a Sturmiansequence, and so a cutting sequence on the torus, by performing a combinatorialoperation called derivation . As noted above, these sequences must have one of theletters isolated; say V ’s are isolated for the sake of example. From each block of UTTING SEQUENCES ON SQUARE-TILED SURFACES 5 consecutive H ’s we then delete one H obtaining a new sequence which is referredto as the derived sequence of the original sequence. If a given sequence is Sturmian,then its derivation will also be Sturmian and so again has an isolated letter. Re-peating the process finitely-many times, the other letter H becomes isolated. Wecan then start the process over again, deleting one letter at a time from blocks ofconsecutive V ’s.Given a Sturmian sequence ε ∈ E Z , we can associate a real number by thefollowing procedure. Suppose that the number of H ’s appearing between adjacent V ’s is a or a + 1 ( a could be zero). Delete a H ’s from each block of consecutive H ’s between a pair of V ’s. The H ’s are now the isolated characters. Now supposebetween two adjacent H ’s there are a or a + 1 V ’s. Again delete blocks of V ’sof size a , leaving one V if there were a + 1 V ’s in the block. Repeat the processalternating between deleting H ’s and V ’s where at each step there are a i or a i + 1 H ’s (respectively, V ’s) between the isolated V ’s (resp., H ’s). Now define the slope of the Sturmian sequence ε to be m = m ( ε ) = a + 1 a + 1 a + 1 a + . . . which is slope of the geodesic on the torus with cutting sequence ε .We will also define the length M ( ε ) of a Sturmian sequence ε as M = M ( ε ) = (cid:22) m ( ε ) (cid:23) , and the rotation parameter θ ( ε ) of ε as θ = θ ( ε ) = 1 m ( ε ) − M ( ε ) . If we consider a circle which is a horizontal line on the torus, the first return mapof the geodesic flow with slope m is precisely a rotation by θ .2.4. Words and languages.
By an alphabet A we mean some finite set of labels.A word with letters in A is a finite, ordered list of elements in A , and the collectionof all words with letters in A is denoted A ∗ . If we define a binary operation on A ∗ by concatenating two words, then A ∗ is the free monoid generated by A .We will use subscripts to denote individual letters of a word w ∈ A ∗ : w is thefirst character of the word, w the second character, and so on. We will denote thelength of a word by | w | .In general, a language with letters in A is simply a subset of A ∗ . We say thata language is minimal if for each word w in the language there exists an integer N such that w is a subword of every word in the language of length at least N .Associated to any language is a dynamical system consisting of all sequences whereevery finite subword of a sequence is a word of the language, together with the shiftmap. A cylinder set [ w ] is the set of all sequences whose first | w | characters givethe word w . These cylinder sets form a base for a topology on the sequence spacemaking it homeomorphic to a Cantor set and for which the shift map is continuous.It is a basic fact that the language is minimal if and only if the associated topologicaldynamical system is minimal: that is, every orbit is dense in the space. Note thatthe cylinder sets form a basis for the topology of this space of sequences, and by CHARLES C. JOHNSON the Kolmogorov extension theorem, a Borel measure on this space is determinedby the measure of each cylinder.Given a biinfinite sequence of letters in A , say u ∈ A Z , the language of u is thecollection of all finite subwords which appear in u . We will say the sequence u is minimal if its language is minimal.2.5. Interval exchange transformations. An interval exchange transformation ,abbreviated IET , is a right-continuous bijection from an interval I to itself which isa piecewise translation having finitely-many discontinuities. That is, an IET cutsthe interval I up into finitely-many pieces and then rearranges the pieces. Such amap is determined by two pieces of information: the combinatorial data of the mapis the permutation describing the rearrangement of the subintervals, and the lengthdata is a vector describing the lengths of the subintervals.We will represent the combinatorial data of a given interval exchange T : I → I as a pair of bijections. If A is a set of k labels, we consider a pair of bijections, π and π , from A to { , , ..., k } so that π − (1) is the label of the left-most subintervalof I before applying T ; π − (1) is the label of the left-most subinterval of I afterapplying T ; π − (2) is the label of the second interval before applying T , and π − (2)is the label of the second interval after applying T ; and so on.We will use the combinatorial data of an interval exchange to define two totalorderings on the symbols of A . For each bijection π : A → { , , ..., d } we define ≤ π as follows: for a, b ∈ A , say a ≤ π b if π ( a ) ≤ π ( b ). By a π -interval of A we mean acollection of consecutive elements of A according to ≤ π .The length data of an interval exchange is given by a vector of lengths ℓ = ( ℓ a ) a ∈A such that each ℓ a > P a ℓ a is the length of the interval I . See Figure 3.D A C BB A D C Figure 3.
The interval exchange with combinatorial data π =( A, B, C, D ) (2 , , , π = ( A, B, C, D ) (2 , , ,
3) andlength data ℓ = (0 . , . , . , . π , π ) is irreducible if there does not exista 1 ≤ j < k such that π − ( { , , ..., j } ) = π − ( { , , ..., j } ) . If the combinatorial data of an interval exchange was not irreducible, then theinterval exchange would decompose into two invariant subintervals for every choiceof length data.
UTTING SEQUENCES ON SQUARE-TILED SURFACES 7
Once the combinatorial and length data are known, the actual operation of themap is easy to describe. For each a ∈ A , let δ a = X b : π ( b ) <π ( a ) ℓ b δ a = X b : π ( b ) <π ( a ) ℓ b and define the subintervals I a = h δ a , δ π − ( π ( a )+1) (cid:17) I a = h δ a , δ π − ( π ( a )+1) (cid:17) . Then the δ a represent the discontinuities of T , and δ a are the discontinuities of T − . The map T acts by translating I a to I a : T ( x ) = x − δ a + δ a if x ∈ I a , and T − ( x ) = x − δ a + δ a if x ∈ I a . We say that an interval exchange T satisfies the infinite distinct orbit condition ,often abbreviated idoc , if for each n > a, b ∈ A with π ( b ) > T n (cid:0) δ a (cid:1) = δ b . It was shown in [Kea75] that the interval exchange T is minimal if and only if itsatisfies the infinite distinct orbit condition.Given a finite word w = w w · · · w n ∈ A ∗ , we will let I w denote the set of pointsinside of I which start off in I w , then proceed to I w when T is applied, then I w when T is applied, and so on: I w = n \ i =0 T − k (cid:0) I w i (cid:1) . Given an interval exchange transformation T and a point x ∈ I , we can considerthe biinfinite sequence of labels obtained by recording the label a when an iterate T n ( x ) is in the interval I a .If we are given a sequence u ∈ A Z , it is natural to ask if u represents thesymbolic trajectory of some point under an interval exchange with intervals labeledby A . Ferenczi and Zamboni give necessary and sufficient conditions in [FZ08]for a sequence of symbols to be the symbolic trajectory of some interval exchangetransformation with fixed combinatorics. Theorem 1 ([FZ08]) . The sequence u ∈ A Z is a symbolic trajectory of an intervalexchange satisfying Keane’s infinite distinct orbit condition with irreducible combi-natorial data π , π : A → { , , ..., k } if and only if the following conditions aresatisfied:(1) u is a minimal sequence;(2) each letter of A appears in u ;(3) for each finite-word w appearing in u , the set of letters which appear asprefixes of w in u forms a π -interval;(4) for each finite-word w appearing in u , the set of letters which appear assuffixes of w in u forms a π -interval; CHARLES C. JOHNSON (5) if a, b ∈ A are admissible prefixes of w with a ≤ π b , and if y is an admissiblesuffix of aw and z an admissible suffix of bw , then y ≤ π z ; and(6) if a and b are admissible prefixes of w , then there is exactly one element of A which is an admissible suffix of both aw and bw . This theorem only says that a given sequence u ∈ A Z is a symbolic trajectoryof some interval exchange with the chosen combinatorial data, but does not tell usif u is a symbolic trajectory for a particular interval exchange with this combina-torial data. Obtaining a particular interval exchange with the given sequence asa symbolic trajectory is equivalent to finding a shift-invariant measure for the dy-namical system associated with the sequence. Given such a measure we can definethe length of the subinterval I a a ··· a n as the measure of the cylinder [ a a · · · a n ].3. Interval Exchanges on Square-Tiled Surfaces
Let Λ = { , , ..., d } be a set of labels, and let h and v be two permutationsof Λ such that h h, v i acts transitively on Λ. Let X be the corresponding square-tiled surface. Denote the geodesic interval at the base of square λ by I λ as inFigure 4(a), and let I = S λ I λ . Now consider the first-return map T m : I → I obtained by firing a geodesic ray from x ∈ I with slope m , and recording wherethe geodesic first intersects I as in Figure 4(b). In Figure 4(b), the base of each I λ is broken into two pieces: a left-hand side and a right-hand side. Geodesics in aleft-hand side of the base of a square stay parallel to one another and are followeduntil reaching the base of another square. The paths which geodesics in a left-handinterval follow are shaded with various textures in Figure 4(b). Similarly, geodesicsemitted from the right-hand side of each base remain parallel and are followed untilreaching the base of another square, and the path followed by these right-handtrajectories are given different shades of grey. I I I I I I (a) The bases of squares form a col-lection of disjoint intervals. (b) We consider the first-return mapon this collection of intervals. Figure 4.
We consider the interval exchanges which are given bythe first-return map on the bases of the squares.In the following we assume that the slope of a geodesic is m >
Lemma 2.
The first-return map on the base of the squares in a square-tiled surfacehas the form T m ( x, λ ) = ( ( R θ ( x ) , vh M ( λ )) if x ∈ [0 , − θ )( R θ ( x ) , vh M +1 ( λ )) if x ∈ [1 − θ, UTTING SEQUENCES ON SQUARE-TILED SURFACES 9 where θ ∈ [0 , and M ∈ { , , , ..., D − } are values depending on m , D dependson the surface, and R θ is the rotation of [0 , by θ .Proof. Since each I λ is a unit interval, we can think of I as [0 , × Λ, identifying I λ with [0 , × { λ } . Suppose that the horizontal permutation h consists of cycles h , h , ..., h k and denote the length of the cycle h i by | h i | . Define D := lcm( | h | , | h | , ..., | h k | ) ,M := (cid:22) m (cid:23) (mod D ) ,θ := 1 m − (cid:22) m (cid:23) , and R θ ( x ) := x + θ − ⌊ x + θ ⌋ . If x ∈ I λ , then the geodesic ray to the North-East emitted from x with slope m cuts across the vertical sides of squares in the same horizontal cylinder as square λ before eventually reaching the base of some square λ ′ in one of the horizontalcylinders connected to the cylinder containing λ . Because the squares are 1 × m , the horizontal distance traveled by the geodesic beforereaching I λ ′ is m .To make the arithmetic simpler, we develop the cylinder containing square λ in the plane, choosing coordinates so that the left-most point of I λ is (0 , (0 ,
0) ( / m , − θ, M M + 1
Figure 5.
Computing the first-return map.The geodesic emitted from x leaves the cylinder at the point ( y,
1) where y := x + 1 m = x + M + θ. When x < − θ , y is in [ M, M + 1), and so cuts across M squares before leaving thecylinder and moving up into square vh M ( λ ). When x ≥ − θ , y ∈ [ M + 1 , M + 2)and so the geodesic leaves the cylinder by moving into square vh M +1 ( λ ).As we identify the base of square λ ′ with the interval [0 , x -coordinate ofour geodesic upon entering I ′ λ is x + 1 m − (cid:22) x + 1 m (cid:23) = R θ ( x ) . The cycles of h determine horizontal cylinders on the surface where the length ofa cylinder equals the length of the corresponding cycle. Given a cycle of length k ,then for each square λ in this cylinder, h M ( λ ) = h M + nk ( λ ) for each n ∈ Z . Hencewe only care about the value of M modulo D . (cid:3) Lemma 3.
The first-return map T m described in Lemma 2 is conjugate to aninterval exchange on [0 , d ) .Proof. Keeping M and θ as defined in Lemma 2, let T be the interval exchange on2 d subintervals of [0 , d ) with labels from A = Λ × { L, R } , combinatorial data π ( λ, s ) = ( λ − s = L λ if s = Rπ ( λ, s ) = ( vh M ( λ ) if s = L vh M +1 ( λ ) − s = R and length data ℓ λ,s = ( − θ if s = Lθ if s = R .
With this choice of combinatorial and length data, the discontinuities of T and T − are respectively δ λ,s ) = ( λ − s = Lλ − θ if s = R and δ λ,s ) = ( vh M ( λ ) − θ if s = Lvh M +1 ( λ ) − s = R. Now consider the map ϕ : [0 , × Λ → [0 , d ) defined by ϕ ( x, λ ) = x + λ − ϕT m = T ϕ . If x ∈ [0 , − θ ), then ϕT m ( x, λ ) = ϕ ( R θ ( x ) , vh M ( λ ))= R θ ( x ) + vh M ( λ ) − x + θ + vh M ( λ ) − x + λ − − ( λ −
1) + vh M ( λ ) − θ = x + λ − − δ λ,L ) + δ λ,L ) = T ( x + λ − T ϕ ( x, λ ) . The proof is similar if x ∈ [1 − θ, (cid:3) In the above we assumed the slope m was positive and the flow was to the North-East just to simplify the statements and proofs of Lemma 2 and Lemma 3. If weconsidered flowing in another direction, then the above arguments still hold except h and/or v may be replaced by their inverses, and R θ may be replaced by R − θ . Forexample, flowing to the North-West with slope m < h, v )is equivalent to flowing to the North-East with slope − m on the surface determinedby ( h − , v ), and similarly for the other directions. UTTING SEQUENCES ON SQUARE-TILED SURFACES 11 Consistent Sequences and Symmetric Sturmian Sequences
Consistent sequences.
Let λ be the label of some square on a square-tiledsurface, and consider the geodesic intervals corresponding to the left-hand andbottom edges of λ , which we will label ( λ, H ) and ( λ, V ), respectively, as in Figure 6.Now consider the geodesic flow with slope m on the surface, where for simplicitywe again consider the case that m > λ it must cross either ( λ, H ) or ( λ, V ), and then must leave thesquare by crossing either ( v ( λ ) , V ) or ( h ( λ ) , H ). The order in which the geodesiccrosses these edges gives a sequence in (Λ × E ) Z . We will adopt the convention thatif the geodesic hits the corner of a square which is not a conical singularity, thentwo symbols are recorded: either ( h ( λ ) , H ) ( vh ( λ ) , V ) or ( v ( λ ) , V ) ( hv ( λ ) , H ).If the upper right-hand corner of square λ is not a conical singularity, then vh ( λ ) = hv ( λ ). The squares λ satisfying this property are called good squares , andsquares with a singularity in the upper right-hand corner are bad squares . (1 ,V ) ( , H ) (2 ,V ) ( , H ) (3 ,V ) ( , H ) (4 ,V ) ( , H ) (5 ,V ) ( , H ) (6 ,V ) ( , H ) Figure 6.
The curves we consider in the cutting sequence.Our basic question is which sequences in (Λ × E ) Z correspond to cutting sequenceson the surface. It is obvious that not all sequences of symbols will be cuttingsequences. The simplest property all cutting sequences must satisfy is what wecall consistency . We say that a sequence ( λ n , ε n ) ∈ (Λ × E ) Z is consistent if λ n +1 = h ( λ n ) when ε n +1 = H , and λ n +1 = v ( λ n ) when ε n +1 = V . To simplifynotation we will let E act on Λ by H · λ = h ( λ ) and V · λ = v ( λ ) with h, v ∈ S Λ fixed permutations.For example, consider the L-shaped surface built from three squares with per-mutations h = (1)(2 3) and v = (1 2)(3) as in Figure 8. A sequence containing ... (1 , H ) (2 , V ) (3 , H ) (2 , H ) ... may be consistent (whether it is in fact consistent or not depends of course on therest of the sequence, but nothing shown here violates consistency), while a sequencecontaining ... (1 , H ) (2 , V ) (1 , H ) (1 , H ) ... can not be consistent because of the (1 , H ) which follows (2 , V ). If the pair ( λ, H )follows (2 , V ) then we must have λ = 3 as h (2) = 3.Since a square-tiled surface X is a cover of the square torus T , geodesics on X project to geodesics on T . Similarly, cutting sequences on X project to cuttingsequences on T . In particular, if ( λ n , ε n ) is a cutting sequence on the square-tiledsurface X , then ε should be a cutting sequence on T . We can try to go in the other direction, lifting a cutting sequence on T to a cutting sequence on X . In particular,given an initial square λ and a cutting sequence ε on T , we can construct a combinatorial lift by defining λ n +1 = ε n +1 · λ n . We may choose λ to be any initialsquare, and so on a square-tiled surface built from d squares there are d differentcombinatorial lifts of each cutting sequence on the torus. Combinatorial lifts areconsistent, but not all consistent sequences are combinatorial lifts. Additionally,every cutting sequence on X is a combinatorial lift of some cutting sequence on T ,but the converse is not true.4.2. A combinatorial lift which is not a cutting sequence.
Consider thecutting sequence associated to the geodesic on the square torus with slope m = = 0 .
422 which passes through the point ( , ,
0) is the lowerleft-hand corner of the square. This geodesic crosses the vertical edge H twicebefore intersecting the top right-hand corner of the square, but then continuesfrom the bottom left-hand corner giving the cutting sequence ε = ...V HHHV... asin Figure 7. Here we made the choice that hitting the corner should correspondto recording HV . We could have just as easily decided to instead record V H toobtain the sequence ε ′ = ...V HHV H... . We will assume that ε and ε ′ are thefirst characters that appear in the strings above; that is, ε = V , ε = H , ε = H , ε = H , ε = V , and ε ′ = V , ε ′ = H , ε ′ = H , ε ′ = V , ε ′ = H , and so on. VVH H
Figure 7.
On the torus, geodesics are still defined even when theyhit the corner.We will now build a consistent sequence which is not the cutting sequence ofa geodesic on a square-tiled surface. We consider the L-shaped surface built fromthree squares which has permutations h = (1)(2 3) and v = (1 2)(3) shown inFigure 8. Now consider the sequences ( λ n , ε n ) and ( λ ′ n , ε ′ n ) on ( { , , } × { H, V } ) Z which are defined by λ = λ ′ = 2 and λ n +1 = ε n +1 · λ n . The sequence λ ′ isconstructed similarly, using ε ′ in place of ε . We then have( λ n , ε n ) = ... (2 , V )(3 , H )(2 , H )(3 , H )(3 , V ) ... ( λ ′ n , ε ′ n ) = ... (2 , V )(3 , H )(2 , H )(1 , V )(1 , H ) ... By construction both of these sequences are consistent with the permutations defin-ing the surface and are combinatorial lifts of the Sturmian sequences ε and ε ′ , butneither sequence is a cutting sequence on this surface. If ( λ n , ε n ) was the cuttingsequence of γ and ( λ ′ n , ε ′ n ) the cutting sequence of γ ′ , then γ and γ ′ would projectto the geodesic in Figure 7. However, no infinitely-long geodesic on this L-shaped UTTING SEQUENCES ON SQUARE-TILED SURFACES 13 surface can project to this geodesic because the geodesic on the torus passes througha corner which is one of the branch points of the cover: on the L-shaped surfacethis is a conical singularity and the geodesic flow ends upon hitting the singularity. ?? ?
Figure 8.
We can easily produce sequences of labels which areconsistent with the gluings of a square-tiled surface, but which donot represent cutting sequences.The choice of λ = 2 is arbitrary in this example. The same phenomenon ofconstructing a cutting sequence which describes a geodesic which intersects a conepoint would happen for any choice of λ on the L-shaped surface since the corner ofeach square is a cone point. On other surfaces where some corners are cone pointsand some are not, however, the choice of initial square λ matters. We will seelater that cutting sequences which describe geodesics passing through a corner mustsatisfy a certain symmetry condition, such a sequence is called almost symmetric below. These almost symmetric sequences are candidates for cutting sequences ofthe torus which do not lift to the square-tiled surface, but some may lift and somemay not, depending on the choice of λ . We will encode the information of whichsequences lift and which do not by considering walks on a special type of graphand will see that there is a way of determining precisely which almost symmetricsequences describe geodesic which intersect a cone point according to properties ofthis walk.Every geodesic on the square torus T which does not pass through the cornerpoint has d , the number of squares, lifts to a square-tiled surface by basic coveringtheory. Thus every cutting sequence of a geodesic on T , which does not describea geodesic passing through the corner, has d combinatorial lifts to the square-tiledsurface. Hence the only serious obstacle to describing cutting sequences on a square-tiled surface is classifying those sequences which describe geodesics passing throughthe corner point.4.3. Symmetric Sturmian sequences.
We will call a Sturmian sequence ε oddsymmetric if there exists an N such that ε N + k = ε N − k , and even symmetric ifthere exists an N such that ε N + k = ε N − k − for every k ≥
0. We will say ε is almost symmetric if there exists an N such that ε N + k = ε N − k − for every k ≥ ε N = ε N − . Below is an example of an odd symmetric sequence, an evensymmetric sequence, and two almost symmetric sequences. The points aroundwhich the sequences are symmetric are highlighted. ...HV V HV V V H V V V HV V...
Odd symmetric ...V V HV V H VV HV V HV V...
Even symmetric ...HV V HV V VH V V HV V V...
Almost symmetric ...HV V HV V HV V V HV V V...
Almost symmetricBy a symmetric
Sturmian sequence we will mean a sequence which is odd sym-metric, even symmetric, or almost symmetric.
Recall that the torus is a hyperelliptic surface and the hyperelliptic involution ι : T → T acts as a 180 ◦ -rotation of the square about the center point. There arefour Weierstrass points fixed by ι indicated in Figure 9. A AA ABBC CD
Figure 9.
The four Weierstrass points on the torus together withgeodesics through those points.
Lemma 4.
A Sturmian sequence is symmetric if and only if it represents the cuttingsequence of a uniformly distributed geodesic which passes through a Weierstrasspoint of the torus. Furthermore,(i) even symmetric Sturmian sequences correspond to geodesics passing throughthe center point D ;(ii) almost symmetric sequences pass through the corner point A .(iii) odd symmetric sequences correspond to geodesics passing through B or C ; and Recall that we are considering uniformly distributed geodesics which must haveirrational slope, and such a geodesic can not pass through two distinct Weierstrasspoints.
Proof of Lemma 4.
Let γ denote the geodesic whose cutting sequence is the Stur-mian sequence ε . Suppose that γ passes through the point P on the side of thesquare labeled ε . Applying ι thus gives a geodesic ι ( γ ) which passes through thepoint ι ( P ) which must also be on the side of the square labeled ε , since ι preservesthe sides of the square. Furthermore, the cutting sequence ι ( γ ) is the same as thecutting sequence of γ , provided we walk along the geodesics γ and ι ( γ ) according tohow those geodesics are oriented. If we had adopted the convention to always walkalong the geodesic in the positive (North-East) direction, or always in the negative(South-West) direction, then the cutting sequences would reverse. When γ passesthrough a Weierstrass point, however, the geodesic γ is preserved and we see thesame cutting sequences, emitted from that Weierstrass point, in both the positiveand negative direction, giving the symmetries as described above. The only oddityhere is to recall that when the Weierstrass point under consideration is the cornerpoint we record the symbols V H , or HV , and this results in the “almost symmet-ric” sequence. This establishes one direction of the lemma, and for the converse weconsider the three cases of even symmetric, odd symmetric, and almost symmetricseparately.In each case we have the following setup: supposing ε is symmetric, there existpoints P and Q on γ ∩ ( H ∪ V ) such that the geodesic emitted from P in thepositive direction has the same cutting sequence as the geodesic emitted from Q in UTTING SEQUENCES ON SQUARE-TILED SURFACES 15 the negative direction. Equivalently, the geodesic emitted from ι ( Q ) in the positivedirection gives the same cutting sequence as the geodesic emitted from P in thepositive direction.(i) Suppose ε is even symmetric, so the ray emitted from P in the positive direc-tion is ε ′ = ε ε ε · · · , and the ray emitted from Q in the negative directionis ε ′ = ε − ε − ε − · · · with ε − = ε , ε − = ε and so on. As the ray emittedfrom P gives ε ≥ while the ray emitted from Q gives ε < , the geodesic γ con-nects Q to P without first intersecting H ∪ V . Additionally, since ε = ε − , P and Q must both be on the same edge, H or V . Since P and Q are involutes,however, the only geodesic that connects them without first passing throughthe sides H or V must pass through the center point D of the square.(ii) Suppose ε is almost symmetric, with the ray emitted from P being ε ≥ andthe ray emitted from Q being ε < − with ε i = ε − i − for i >
0, and ε − ε beingone of HV or V H .The portion of the geodesic from Q to P is responsible for the HV (or V H )at ε − ε . Suppose we followed γ backwards from P to some point R on thevertical edge H . We would likewise follow γ forwards from Q to ι ( R ) whichwould also be on H . If R = ι ( R ), then this would produce two distinct H ’s,which do not appear at positions ε − and ε in our sequence. Similarly, if R were on the horizontal edge V , then ι ( R ) would be as well, and if R = ι ( R )we would have two distinct V ’s which do not appear at positions ε − and ε in the sequence. We must then have that R = ι ( R ). If R was either of theWeierstrass points labeled B or C in Figure 9 we would thus produce a singlecharacter in our sequence, but not the two characters we lack.Since our sequence contains V H (or HV ) and intersecting a non-Weierstrasspoint would produce HH or V V , and a Weierstrass point which is not A wouldproduce a since H or single V , the only remaining option is for the geodesicto pass through the Weierstrass point labeled A at the corner of the squareto produce the V H (or HV ).(iii) If ε is odd symmetric, then we may choose P and Q such that the ray emittedfrom P gives ε > while the ray emitted from Q gives ε < with ε i = ε − i for i >
0. This again implies that P and Q are both on H or both on V .Following γ from Q to P crosses the edge ε , while following γ backwardsfrom P to Q crosses ε at the same point. Since Q and P are involutes, thispoint on ε is fixed, and so is either A , B , or C . The point can not be A ,however, as geodesics through A give almost symmetric sequences, and oursequence is not almost symmetric. Thus an odd symmetric sequence mustpass through either B or C . (cid:3) Recall that given a Sturmian sequence ε , its derived sequence is the sequence ε ′ obtained from ε by deleting one character from each block of the non-isolated char-acter. Geometrically, this derivation corresponds to applying the vertical, down-ward Dehn twist when V is isolated, or the horizontal, left-ward Dehn twist when H is isolated. This alters the direction of a geodesic, and the derived sequence issimply the cutting sequence of this new geodesic. Lemma 5.
If a Sturmian sequence ε is almost symmetric, then so is its derivation ε ′ . Proof. If ε is almost symmetric, then it is the cutting sequence of a geodesic throughthe corner of the square. Since the corners are preserved by the Dehn twists above,the derivation ε ′ also corresponds to a geodesic through the corner and so is almostsymmetric as well. (cid:3) Characterizing Cutting Sequences
In this section we prove a characterization theorem for cutting sequences onsquare-tiled surfaces. We will first make note of certain necessary conditions sucha cutting sequence must satisfy, and then convert a sequence satisfying those con-ditions into a symbolic trajectory for an interval exchange and verify the Ferenczi-Zamboni conditions. We then show that there exists a shift-invariant measureagreeing with the lengths of intervals on our square-tiled surface to conclude thatthe sequence is a symbolic trajectory for the interval exchange on our surface.5.1.
Necessary conditions.
Recall that a sequence ( λ n , ε n ) in letters Λ × E is consistent if λ n +1 = ε n +1 · λ n and is a combinatorial lift if ε n is the cutting sequenceof a geodesic on T . Obviously being a combinatorial lift is a necessary conditionfor ( λ n , ε n ) to be a cutting sequence, but as the example in Section 4.2 shows it isin general not a sufficient condition.We will say that ( λ n , ε n ) is almost symmetric around a bad square if ε n is analmost symmetric sequence, so ε N + k = ε N − k − for some N and all k ≥ ε N = ε N − , and λ N is a bad square.5.2. The periodic case.
If ( λ n , ε n ) is a periodic combinatorial lift, then ε n mustbe periodic. Periodic cutting sequences on the torus correspond to geodesics ofrational slope on the torus. Any lift of such a geodesic to the square-tiled surfacewith permutations h and v is a periodic geodesic as well by basic lifting propertiesof covering spaces and the fact that the square-tiled surface and torus are locallyisometric.5.3. The non-symmetric case.
Combinatorial lifts of Sturmian sequences whichare not almost symmetric, and so do not represent cutting sequences passingthrough a corner of the square, easily lift to cutting sequences on every square-tiled surface.It is clear, however, that there may be geodesics on a square-tiled surface whoseprojection to the torus gives an almost symmetric cutting sequence. This canhappen if the geodesic goes through the corner of a good square. To determinewhen this happens we will associate a family of graphs to the surface which describepossible transitions between the squares when the geodesic leaves one horizontalcylinder for another.5.4.
The Γ M graphs. In the description of T m from Section 3, the left-hand subin-terval of I λ always maps to the right-hand side of some I λ ′ and similarly the right-hand subinterval maps to the left-hand side of some I λ ′′ where λ ′′ = vhv − ( λ ′ ).This is true for every slope m , and so the only part of the combinatorial data of theinterval exchange that may change is the number of squares to cross over beforea point on the base exits the top of a cylinder. See Figure 5 on page 9. Let M denote the number of vertical edges (labeled H ) crossed when transitioning fromthe left-hand side of I λ to the right-hand side of I λ ′ In Figure 5, for example, M = 2. In terms of Sturmian sequences, M is the minimal number of H ’s between UTTING SEQUENCES ON SQUARE-TILED SURFACES 17 two V ’s in the cutting sequence, regardless of which letter is isolated. The value of M is determined by the slope m , and considering that the line emitted from (0 , m intersects the line y = 1 at the point ( / m , M = (cid:4) m (cid:5) .Notice that if V ’s are isolated (meaning m < H ’s between V ’s have length M or M + 1. If H ’s are isolated ( m >
1, and so M = 0), then thenumber of H ’s between V ’s is either 0 or 1. As blocks of consecutive H ’s may havelength M or M + 1, we will refer to M as the length of the Sturmian sequence.The interval exchange T m is completely determined by the slope m ; in particular,writing m = M + θ , the value of θ determines the length data of T m and the valueof M determines the combinatorial data. Representing this combinatorial data asa pair of permutations ( π , π ) for a given T m the permutation π is the same forall values of M while π may change. For this reason we will write π M to mean thepermutation associated to the interval exchange T m when (cid:4) m (cid:5) = M .The combinatorial information of ( π , π M ) can be described as a graph Γ M =Γ Mh,v . This will be a labeled, directed multigraph with vertex set Λ, each vertexhaving two outgoing edges labeled L and R , and two incoming edges labeled L and R . We will call such a multigraph a . The edges from vertex λ ingraph Γ M are given by vh M ( λ ) L ←−−−− λ R −−−−−→ vh M +1 ( λ ) . If λ belongs to a cycle of h with length ℓ , then h M ( λ ) = h M + kℓ for all k . Letting h , h , ..., h p denote the cycles of h , and | h i | the length of cycle h i , there are only D = lcm( | h | , | h | , ..., | h p | )possible choices of combinatorial data ( π , π M ). To see this, notice that if M > D ,say M = pD + r for some 0 ≤ r < M , then for each cycle h i we have h Mi = h pD + ri = (cid:0) h Di (cid:1) p · h ri = h ri since h Di = id as D is a multiple of the length of cycle h i . This is true for eachcycle h i and so h M = h r . As T m takes points on the left-hand side of the base ofsquare λ to square vh M ( λ ), and points on the right-hand side of the base of square λ to square vh M +1 ( λ ), the value of M only matters modulo D , and so there areonly D choices for the combinatorial data of T m .As an example, consider the flow on the surface of Figure 1 in a direction withslope m in the interval ( / , / ). For such a slope we will have M = (cid:4) m (cid:5) = 2 andso consider the graph Γ where for the vertex labeled λ has an outgoing edge to vh ( λ ) labeled L , and an outgoing edge to vh ( λ ) labeled R as in Figure 10.Each edge of this graph represents a possible transition in symbolic trajectories of T m : the edge labeled L out of λ and ending at vh M ( λ ) corresponds to points in the[0 , − θ )-portion of I λ being mapped to I vh M ( λ ) , and similarly for the edge labeled R . Hence every symbolic trajectory of T m determines an infinite walk on Γ M . Wewill relate cutting sequences to symbolic trajectories of the interval exchanges, andso infinite walks on these graphs.5.5. Combinatorial lifts and symbolic trajectories.
Let ( λ n , ε n ) be a com-binatorial lift of some Sturmian sequence. We will convert ( λ n , ε n ) to a sequencewhich we will show is the symbolic trajectory of an interval exchange on the square-tiled surface. First we define a subsequence µ i = λ n i of ( λ n ) by deleting those λ n LRL RLRL RLR L R
Figure 10.
The graph Γ associated to the square-tiled surfacewith permutations h = (1)(2 3 4)(5 6) and v = (1 2)(3 5)(4 6).where ε n = H . Now define a sequence ( σ i ) ∈ Σ Z = { L, R } Z by considering thedistance from λ n i and λ n i +1 : σ i = ( L if n i +1 − n i = M + 1 R if n i +1 − n i = M + 2where M is the length (i.e., minimal number of H ’s between two V ’s – this may bezero) of the Sturmian sequence ε .We claim this new sequence ( µ n , σ n ) ∈ (Λ × Σ) Z is a symbolic trajectory for aninterval exchange with combinatorial data ( π , π M ). To see this we need to verifythe Ferenczi-Zamboni conditions. We will establish this through a series of lemmasdescribing properties of ( µ n , σ n ) and the graph Γ M which encodes the combinatorialdata. Lemma 6. If ( λ n , ε n ) is a combinatorial lift of a Sturmian sequence ε , then thesequence σ defined above is a Sturmian sequence in the letters L and R . If ε isalmost symmetric, then so is σ .Proof. Let γ be the geodesic on the square torus represented by the Sturmiansequence ε , let I be the interval at the base of this square, and R : I → I thecircle rotation (interval exchange transformation) given by the first-return map tothe base I of the square under the geodesic flow in the direction of γ . This is aninterval exchange on two intervals; we label the first, left-hand interval L , and thesecond, right-hand interval R . The sequence σ is precisely the sequences of L ’s and R ’s obtained by walking along γ and recording an L when γ intersects the left-handinterval and an R when γ intersects the right-hand interval. Since this sequenceof L ’s and R ’s records the orbit of a point under an irrational rotation (irrationalsince ε is Sturmian hence the slope of the geodesic is not a rational number), σ isitself a Sturmian sequence. See Figure 11 which shows the two different, parallelgeodesic segments. (The picture is drawn on a portion of the universal cover of thetorus to make the picture easier to understand.) Notice that the geodesic γ L whichpasses through the left-hand interval at the base of the first square intersects twovertical lines, while the geodesic γ R which passes through the right-hand intervalcrosses three vertical lines. That is, the corresponding cutting sequence is V HHV for γ L and V HHHV for γ R . In constructing σ we are simply using the number of H ’s (vertical line segments crossed) to determine if the geodesic previously passedthrough a left- or right-hand interval on the base of the square. UTTING SEQUENCES ON SQUARE-TILED SURFACES 19 γ L γ R L R
Figure 11.
The distance between two consecutive V ’s in thecutting sequence of a geodesic determines, and is determined by,whether the geodesic enters the left- or right-hand side of the baseof the square.If V is the isolated character in an almost symmetric Sturmian sequence ε , thenthe sequence has the form ωHV M V HV M Hω or ωHV M HV V M Hω. with ω being the same sequence as ω , but in reverse order. The constructed sequence σ thus has the form τ RL M RLL M Rτ or τ RL M LRL M Rτ. where τ is obtained by applying the rule for determining σ i above to the sequence ω . Similarly, if ε is an almost symmetric Sturmian sequence but with isolated char-acter V , then ε has the form ωV H M V HH M V ω or ωV H M HV H M V ω, and the corresponding σ sequence has the form τ LRτ or τ RLτ. In either case, we exactly have that σ is almost symmetric. (cid:3) Lemma 7.
Each Γ M graph is strongly connected.Proof. As the edges from vertex λ go to vh M ( λ ) and vh M +1 ( λ ), we must show thatthere is some composition of powers of vh M and vh M +1 which sends any given λ i to any λ j . Since h h, v i acts transitively on Λ, it suffices to show that h and v canbe formed by compositions of powers of vh M and vh M +1 .Since Λ has finitely-many elements, there exists some p > vh M ) p =id and so ( vh M ) p − = ( vh M ) − . Thus( vh M ) p − vh M +1 = h − M v − vh M +1 = h. Similarly, there exists a q such that h q = h − and so vh M (( vh M ) p − vh M +1 ) qM = vh M ( h q ) M = vh M h − M = v. (cid:3) Corollary 8.
The permutations ( π , π M ) are irreducible. Proof.
If the permutations were not irreducible, then the corresponding intervalexchange on the surface would have a proper subset of intervals which are leftinvariant by the interval exchange. These invariant subsets of intervals would cor-respond to connected components of the graph Γ M , but Lemma 7 shows there isonly one such component. (cid:3) We will say that an edge of Γ M is a bad edge if traversing that edge couldcorrespond to hitting a conical singularity on the square-tiled surface. Each badsquare of the surface gives two bad edges on each Γ M : if λ is a bad square, thenthe edge labeled L with target v ( λ ) and the edge labeled R with target vh ( λ ) arebad edges.Notice that ( µ n , σ n ) n ∈ Z determines a walk on the graph Γ M , where µ n +1 isobtained by traveling from µ n along edge σ n . We will say that the walk ( µ n , σ n ) is almost symmetric about a bad edge if there exists an N such that σ N + k = σ N − k − for each k ≥ µ N σ N −−−−−→ µ N +1 is a bad edge of Γ M .We now claim that Sturmian walks , walks where the sequence of crossed edgesform a Sturmian sequence, on a Γ M graph are in one-to-one correspondence withsymbolic trajectories of interval exchanges T m on the corresponding square-tiledsurface.Notice that symbolic trajectories of the interval exchange on the bases of squaresin a square-tiled surface give Sturmian walks on the corresponding Γ M graph. Thisis simply because the sequences of edges labeled L and R coincide with the momentswhen the corresponding geodesic on the torus intersects the left- and right-handintervals on the base of the square. As we are considering irrational directions,these L and R subintervals are given by an irrational rotation, and so producea Sturmian sequence. To show that each such Sturmian walk corresponds to asymbolic trajectory of the interval exchange we need to verify the Ferenczi-Zamboniconditions. We first introduce some notation for the admissible prefixes and suffixesof a word.Given a word w = ( ν , s ) ( ν , s ) · · · ( ν k , s k ) ∈ (Λ × Σ) ∗ , let ( p n ( w )) n ∈ Z be thesequence which gives the locations of w in the sequence ( µ n , σ n ) ∈ (Λ × Σ) Z . Thatis, p n ( w ) satisfies ( µ p n ( w ) , σ p n ( w ) ) = ( ν , s ) , ( µ p n ( w )+1 , σ p n ( w ) ) = ( ν , s ) , ...( µ p n ( w )+ k , σ p n ( w ) ) = ( ν k , s k ) . For example, suppose Λ = { , } and consider the word w = (1 , L ) (2 , R ) (1 , L ) (2 , R ) . If the sequence ( µ n , σ n ) contained ... (2 , L ) (1 , L ) (2 , R ) (1 , L ) (2 , R ) (1 , L ) (2 , R ) (1 , L ) (2 , R ) (1 , R ) ... UTTING SEQUENCES ON SQUARE-TILED SURFACES 21 with the initial (2 , L ) corresponding to ( µ , σ ), then we would have p ( w ) = 1, p ( w ) = 3, p ( w ) = 5. Notice that we allow different copies of w to overlap oneanother in our definition of p n ( w ).We will adopt the convention that p ( w ) is the first occurrence of w starting atan index n ≥ µ n , σ n ), though this particular choice of starting point will notaffect what is to follow.Following the notation of [FZ08], we let A ( w ) denote the set of characters whichoccur immediately before w appears in ( µ, σ ), and D ( w ) denotes the set of charac-ters which occur immediately after w : A ( w ) = (cid:8) ( ℓ, s ) (cid:12)(cid:12) ℓ = µ p n ( w ) − , s = σ p n ( w ) − for some n (cid:9) and D ( w ) = (cid:8) ( ℓ, s ) (cid:12)(cid:12) ℓ = µ p n ( w )+ k +1 , s = σ p n ( w )+ k +1 for some n (cid:9) . In the example above we would thus have (2 , L ) , (2 , R ) ∈ A ( w ) and (1 , L ) , (1 , R ) ∈ D ( w ). In principle the set of prefixes and suffixes, which were called arrival and departure sets in [FZ08], for a general language could be quite large and compli-cated sets of letters. The following lemma shows that these sets are actually verysimple for the languages we are considering. Lemma 9.
The sets A ( w ) and D ( w ) each contain at most two characters, A ( w ) is a π M -interval, and D ( w ) is a π -interval.Proof. It is clear that A ( w ) has two characters since there are only two edges inΓ M ending at ν : one labeled L and one labeled R . From the definition of thegraph, we can see that the edge labeled L ending at ν starts at vertex h − M v − ( ν )and the edge labeled R ending at ν starts at vertex h − ( M +1) v − ( ν ). It is alsoeasy to see from the definition of the permutation π M that these two symbols,( h − M v − ( ν ) , L ) and ( h − ( M +1) v − ( ν ) , R ), are π M -consecutive: π M ( h − M v − ( ν ) , L ) = 2 vh M h − M v − ( ν ) = 2 ν π M ( h − ( M +1) v − ( ν ) , R ) = 2 vh M +1 h − ( M +1) v − ( ν ) = 2 ν − . The proof that D ( w ) is a π -interval and that it contains at most two elements issimilar. (cid:3) Lemma 10. If ( σ n ) is a Sturmian sequence in { L, R } , any ( µ n , σ n ) satisfying µ n +1 = σ n · µ n is minimal.Proof. We need to show that for each finite word of ( µ n , σ n ), there exists somebound M such that every word of length M has the given finite word as a factor.Shifting if necessary, we may assume the finite word is( µ , σ ) ( µ , σ ) · · · ( µ k , σ k ) . As ( σ n ) is a Sturmian sequence, there exists some N such that all words of ( σ n )of length N have σ σ · · · σ k as a factor. Letting ( τ i ) be the sequence of integerssatisfying σ τ i = σ , σ τ i = σ , · · · , σ k + τ i = σ k , with τ = 0, the ( τ i ) sequence has bounded gaps: τ i +1 − τ i < N . To showminimality of ( µ n , σ n ), it suffices to show that there exists an M such that µ τ i = µ for some i < M . To show this it will be helpful to think about the sequence ( µ n , σ n )as determining a walk on a graph Γ M . Let ψ be the empty word and set ψ i = σ τ i − σ τ i − · · · σ τ i − . We can interpretthe ψ i ’s both as words of ( σ n ), and as permutations of Λ with ψ being the identity.Notice µ τ = ψ ( µ ) and µ τ i = ψ i ( µ τ i − ) = ψ i ψ i − · · · ψ ( µ ).The first few edges crossed by the walk ( µ, σ ) are as follows: µ µ · · · µ τ σ σ σ τ − ψ where the walk from µ to µ τ has at most N characters. If µ τ = µ , then considera longer portion of the walk until ψ appears again, ending at some µ τ i . Since( σ n ) is Sturmian, there exists some N so that all words of length N have ψ asa factor. Thus the distance from µ to µ τ i is at most N + N . If µ does notappear in µ τ , µ τ , ..., µ τ i , notice that µ τ i can not equal µ (by assumption), butcan also not equal µ τ since µ τ i = ψ ( µ τ ( i − ) and µ τ ( i − = µ . µ · · · µ τ · · · µ τ ( i − · · · µ τ i ψ ψ ρ Let ρ be the product of all the symbols (permutations) between µ and µ τ i . As( σ n ) is Sturmian, there exists an N such that all words of length N contain ρ asa factor. Consider now the shortest portion of our walk which contains a secondfactor of ρ and ends at some µ τ i . µ · · · µ τ i · · · µ τ ( i − · · · µ τ i ρ ρ ρ If no µ through µ τ i equals µ , then µ τ i is not equal to µ , µ τ , or µ τ i , and thenumber of steps from µ to µ τ i is at most N + N + N . Letting ρ denote thetransitions from µ to µ τ i , there exists some N such that all words of length atleast N contain a factor of ρ , and we can consider the shortest portion of ourwalk starting from µ and containing two disjoint factors of ρ .We continue in this way building a sequence of symbols µ τ ik which can not takeon k + 1 different symbols of Λ. If we repeat this procedure up until µ τ id , then thissymbol can not take on any value in the alphabet. Hence at some point µ mustrepeat, and this must happen within N + N + N + · · · + N d steps. (cid:3) Corollary 11. If ( λ n , ε n ) is a combinatorial lift of a Sturmian sequence, then theassociated sequence ( µ n , σ n ) is minimal. Lemma 12. If ( σ n ) is a Sturmian sequence in Σ Z , each ( µ n , σ n ) satisfying µ n +1 = σ n · µ n contains every element of Λ × Σ .Proof. Thinking of ( µ n , σ n ) as a walk on a graph Γ = Γ M , the lemma claims thatevery edge of the graph is crossed by the walk. Suppose this was not the caseand there was some edge the walk does not cross. We will perform an iterativeprocedure replacing our graph with a derived graph Γ ′ so that the uncrossed edge UTTING SEQUENCES ON SQUARE-TILED SURFACES 23 of Γ becomes a vertex which the corresponding derived walk ( µ ′ n , σ ′ n ) avoids. Thederived graph will be a 2-oriented graph with the same number of vertices and edgesas Γ, and so a walk which avoids one vertex implies there are at least three edgesof Γ ′ which the derived walk does not cross. Repeating the derivation procedureproduces a graph Γ ′′ and a walk ( µ ′′ n , σ ′′ n ) which avoids two vertices, and at leastfour edges are not crossed. Since our graphs all have the same finite number ofvertices and edges, this procedure will after finitely-many steps show that our walkmust not cross any edge.We define the derived graph Γ ′ and derived walk ( µ ′ n , σ ′ n ) as follows. Supposethat the edge of Γ which is not crossed is λ σ −→ λ ′ . Let τ denote the element of Σ which is not the σ above. Now consider blocks of( σ n ) which consist of blocks of consecutive τ ’s and end in σ . There will be two suchblocks, but the possible blocks depend on the length M of the Sturmian sequenceand whether σ or τ is the isolated character. We will refer to the two possibleblocks as L ′ and R ′ .If σ is the isolated character of ( σ n ), then the blocks ending in σ have the form L ′ = τ M σ or R ′ = τ M +1 σ. If τ is the isolated character, then the blocks have the form L ′ = σ or R ′ = τ σ. In either case we may interpret L ′ and R ′ as permutations of Λ and consider thegraph Γ ′ with vertices the characters of Λ and edges L ′ ( λ ) L ′ ←− λ R ′ −→ R ′ ( λ ) . Notice that Γ ′ is a 2-oriented graph with the same number of edges and verticesas Γ. Furthermore, the proof of Lemma 7 applies equally well to Γ ′ , and so Γ ′ isstrongly connected.The derived walk ( µ ′ n , σ ′ n ) consists of the sequence σ ′ n of L ′ and R ′ obtainedby replacing the blocks in ( σ n ) above with L ′ and R ′ , and setting µ ′ = µ and µ ′ n +1 = σ ′ n · µ ′ n .Notice that ( σ ′ n ) is a Sturmian sequence. In the case that τ is the isolatedcharacter, replacing blocks of τ σ with R ′ and blocks of σ with L ′ is the same asderiving the Sturmian sequence (removing a single σ from blocks of consecutive σ ’s), which gives a Sturmian sequence, and then doing a character-by-characterreplacement by substituting τ with R ′ and σ with L ′ , which is obviously a Sturmiansequence. In the case that σ is the isolated character, ( σ ′ n ) is obtained by deriving( σ n ) M times so that τ becomes the isolated character, and then performing theoperation previously described.The derived walk ( µ ′ n , σ ′ n ) on Γ ′ is an acceleration of the walk ( µ n , σ n ) on Γwhere we consider longer pieces of the original walk ending in σ . Since λ σ −→ λ ′ isnot crossed in the original walk, there are no blocks crossing such an edge. Sincethe other incoming edge around λ ′ must be labeled by τ , the derived walk avoidsthe vertex λ ′ of Γ ′ . As there are two edges which enter λ ′ , the derived walk mustnot cross either of these edges, and can not cross either of the outgoing edges either.It may happen that one of the incoming edges may also be an outgoing edge (i.e.,there may be a loop at λ ′ ). Thus there are at least three forbidden edges of Γ ′ the derived walk can not cross. There can not be fewer than three edges because thiswould violate the strong connectedness of the graph.As ( σ ′ n ) is a Sturmian sequence, one character, say σ ′ , is isolated and we canrepeat the above procedure by considering blocks of ( σ ′ n ) which end in σ ′ to producea second derived graph Γ ′′ and second derived walk ( µ ′′ n , σ ′′ n ). Again, Γ ′′ will be astrongly-connected 2-oriented graph and ( σ ′′ n ) will be a Sturmian sequence. As( σ ′ n ) avoids two edges, the walk ( µ ′′ n , σ ′′ n ) avoids at least four edges. Loops mayoccur and so the number of avoided edges does not double each time. However, thestrong connectedness of the graph prevents us from having a connected componentof avoided edges and so the number of forbidden edges must grow by at least one.Each time we replace a graph and a walk on that graph with its derivation inthis manner, the number of avoided edges grows by at least one. Since there areonly finitely-many edges, however, after finitely-many derivations, our walk, whichis given by a Sturmian sequence, can not cross any edge of the graph. Thus therecan not be any edge avoided by the graph, and so the original sequence ( µ n , σ n )must contain each element of Λ × Σ. (cid:3) Proposition 13.
A Sturmian walk on Γ M satisfies the Ferenczi-Zamboni condi-tions and thus is the symbolic trajectory of some interval exchange with combina-torial data ( π , π M ) .Proof. Let ( µ n , σ n ) ∈ (Λ × Σ) Z denote the walk. We simply verify that the sixconditions of Theorem 1 are satisfied. Condition (1) is ensured by Lemma 10.Condition (2) is given by Lemma 12. Conditions (3) and (4) are given by Lemma 9.For condition (5), we remark that since σ is a Sturmian sequence it representsthe symbolic trajectory of a point under an irrational rotation and so must satisfythe Ferenczi-Zamboni conditions. In particular, condition (5) holds on the factor σ in our sequence since this is a Sturmian sequence: σ is the symbolic trajectory ofan interval exchange with intervals labeled by Σ = { L, R } with combinatorial data π ( L ) = 1 π ( L ) = 2 π ( R ) = 2 π ( R ) = 1and length data ℓ L = 1 − θ , ℓ R = θ where θ is the rotation parameter describedearlier in Section 2.3.Now suppose that w = ( ν , s ) ( ν , s ) · · · ( ν k , s k )is some finite subword of ( µ, σ ). Let s denote the Sturmian word s s · · · s k . Theadmissible prefixes and suffixes of w are then completely determined by the admis-sible prefixes and suffixes of s . In particular, there are only two points of concernin showing condition (5) holds, and both of these concerns are alleviated in light ofthis observation.1. Suppose A ( w ) is a singleton, say ( ξ, t ) is the unique element of A ( w ). We mustshow that D (( ξ, t ) w ) is a singleton as well. However, if A (( ξ, t ) w ) is a singleton,then so is A ( t s ) and since the Ferenczi-Zamboni conditions hold for the sequence( s n ), D ( t s ) must be a singleton. This then implies that D (( ξ, t ) w ) is a singleton,since the second component must be the only element in D ( ts ) and the first oneis determined by µ n +1 = σ n µ n . (Note here that the inequalities in condition (5)of the Ferenczi-Zamboni conditions occur simultaneously.) UTTING SEQUENCES ON SQUARE-TILED SURFACES 25
2. Suppose now that A ( w ) = (cid:8) ( R − ( ν ) , R ) , ( L − ( ν ) , L ) (cid:9) , where R − = h − ( M +1) v − and L − = h − M v − . Then A ( s ) = { R, L } . As( R − ( ν ) , R ) ≤ π ( L − ( ν ) , L ) and ( s k · ν k , L ) ≤ π ( s k · ν k , R ), we must showthat if ( s k · ν k , R ) ∈ D (( R − ( ν ) , R ) w ), then D (( L − ( ν ) , L ) w ) = { ( s k · ν k , R ) } .However, since the Sturmian sequence ( s n ) n ∈ Z satisfies the Ferenczi-Zamboniconditions, this follows immediately from the fact that if A ( s ) = { L, R } and R ∈ D ( R s ) then D ( L s ) = { R } .To show condition (6), we again remark that the Sturmian sequence σ satisfiesthe Ferenczi-Zamboni conditions. In particular, condition (6) in the case of σ implies that if w is a finite subword of σ , and s, s ′ ∈ A ( w ) are distinct symbols,then D ( sw ) ∩ D ( s ′ w ) must be a singleton. This trivially implies the same for thesequence ( µ, σ ). (cid:3) We are now able to combine the results above to establish our main theorem,restated below for the convenience of the reader.
Theorem 14 (The Characterization Theorem) . Let X be a square-tiled surface on d squares determined by a pair of permutations h and v on Λ = { , , ..., d } . Let E = { H, V } be symbols for the edges of the unit square torus. Label the left-handedge of square λ as ( λ, H ) and the bottom edge as ( λ, V ) . Then a biinfinite sequence ( λ n , ε n ) ∈ (Λ × E ) Z is the cutting sequence of an infinite geodesic on X if and onlyif the following conditions are satisfied:(1) ( λ n , ε n ) is consistent with the gluings of the surface;(2) ( λ n , ε n ) is either periodic, or is minimal but not almost symmetric arounda bad square of the surface; and(3) ε n is the cutting sequence of a geodesic on the square torus. The strategy of the proof will be to take a sequence satisfying the three condi-tions of Theorem 14 and associate to that sequence the symbolic trajectory of aninterval exchange on the surface. Such an association necessarily loses some infor-mation as two different geodesics with different cutting sequences could give rise tothe same symbolic trajectory if those geodesics are related by a Dehn twist. Thecorrespondence between cutting sequences and symbolic trajectories is not one-to-one, but instead depends on a choice which essentially keeps track of how manytimes Dehn twists were applied.
Proof of Theorem 14.
One direction of the proof is simple: it is clear that a cuttingsequence ( λ n , ε n ) of a geodesic γ on the surface satisfies the three conditions sincethe geodesic on X projects to a geodesic on the square torus T . By the Veechdichotomy, the geodesic flow in any given direction on X or T is either periodic(if the flow is in a rational direction), or uniquely ergodic (if the flow is in anirrational direction). As Lebesgue measure is preserved by flows in any direction,if the flow is uniquely ergodic the flow must in fact be minimal as it will enter anyopen ball on the torus. Since the flow on any translation surface is the suspensionover an interval exchange (coming from the first-return map of the geodesic flow toan appropriately chosen transversal geodesic segment), minimality of the flow onthe surface implies minimality of the interval exchange which implies minimality of symbolic trajectories. Thus non-periodic cutting sequences must come fromgeodesics in irrational directions and so are required to be minimal.For the converse, suppose from ( λ n , ε n ) we construct sequences ( µ n , σ n ) as above.Then by Proposition 13, ( µ n , σ n ) is the symbolic trajectory of some interval ex-change with the combinatorial data ( π , π M ) as determined by the surface. Weneed to show now that the interval exchange on the bases of the squares, as de-scribed in Section 3, is one of the interval exchanges for which ( µ n , σ n ) is a symbolictrajectory. By the proof of [FZ08, Theorem 2], this means we need to exhibit ashift-invariant measure on the dynamical system described by ( µ n , σ n ) where themeasure on each cylinder set agrees with the lengths of the subintervals of theinterval exchange on the surface.We consider the interval exchange T m : I × Λ → I × Λ corresponding to the flowwith slope m being the slope of the Sturmian sequence ε as described in Section 2.3.Define a measure µ on the associated shift space by declaring the measure ofa cylinder set to be the Lebesgue measure of the corresponding subinterval in ourinterval exchange. Denoting the length of an interval J by | J | , we are thus consid-ering the measure given by µ ([ w ]) = | I w | . This clearly defines a measure, but wemust check that the measure is shift invariant.Let ∆ denote the shift, and let T denote the interval exchange on the surface.We need to show that for each cylinder [ w ], µ ([ w ]) = µ (∆ − ([ w ])). Notice∆ − ([ w ]) = [ a ∈ A ( w ) [ aw ]where again A ( w ) is the set of allowable one-character prefixes of the word w .We must show the length (cid:12)(cid:12) I w (cid:12)(cid:12) of the subinterval I w equals the measure of the∆-preimage of w .We may express the measure of the preimage of w as µ [ a ∈ A ( w ) [ aw ] = X a ∈ A ( w ) (cid:12)(cid:12) I aw (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ a ∈ A ( w ) (cid:2) I a ∩ T − ( I w ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ a ∈ A ( w ) I a ∩ T − ( I w ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T [ a ∈ A ( w ) I a ∩ T − ( I w ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ a ∈ A ( w ) I a ∩ I w (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By the assumption that the Ferenczi-Zamboni conditions are satisfied, A ( w ) isa π M -interval, and so S a ∈ A ( w ) I a is a subinterval of I . Since elements of A ( w )are precisely the a such that I aw = ∅ , the interval S a ∈ A ( w ) I a contains I w which UTTING SEQUENCES ON SQUARE-TILED SURFACES 27 implies shift-invariance of our measure µ . Thus T m is one of the interval exchangesfor which the given sequence is a symbolic trajectory.For each choice of M ∈ { , , , ..., D } , with D the least common multiple of thelengths of cycles of the permutation h , and each choice of k ∈ N , there is a one-to-one correspondence between symbolic trajectories of the interval exchange on oursurface with combinatorial data ( π , π M ) and cutting sequences ( λ n , ε n ) where thelength modulo D of the Sturmian sequence ( ε n ) is M .To see this, we simply replace the characters in the sequence as follows: ( µ, L )is replaced by ( µ, V ) ( h ( µ ) , H ) ( h ( µ ) , H ) · · · ( h kD + M ( µ ) , H ) , and ( µ, R ) is replaced by( µ, V ) ( h ( µ ) , H ) ( h ( µ ) , H ) · · · ( h kD + M ( µ ) , H ) ( h kD + M +1 ( µ ) , H ) . The values of M and k have a very geometric interpretation. Given any geodesic γ that crosses the base of a square λ on the surface, consider the geodesic δ whichis obtained by applying a multiple Dehn twist to the surface so that each horizontalcylinder formed by the squares tiling the surface (i.e., corresponding to each cycleof the permutation h ) has its top and bottom edges fixed. The geodesics γ and δ certainly have different cutting sequences since δ will cross more vertical edgeslabeled H than γ will. However, both geodesics have the same symbolic trajectoryunder the interval exchange on the bases of the squares. See Figure 12 for anexample. Figure 12.
Though γ (solid) and δ (dashed) give the same sym-bolic trajectory of the interval exchange on the surface, they havedifferent cutting sequences.That is, the sequence ( µ n , σ n ) does not contain enough information to determinea cutting sequence on the surface since two different geodesics, related by an ap-propriate Dehn twist, will have different cutting sequences but the same ( µ n , σ n )sequence. The k tells us how many times the Dehn twists are applied and arenecessary for recovering the cutting sequence from the symbolic trajectory of theinterval exchange.Finally, note that a sequence which is symmetric about a bad square correspondsto a symbolic trajectory of a discontinuity of the interval exchange. (cid:3) Concluding Remarks
We end by stating one simple consequence of Theorem 14. If the permutations h and v defining the square-tiled surface are such that h ( λ ) = v ( λ ) for each λ ∈ Λ,then each cutting sequence is uniquely determined by the sequence of labels of squares, ( λ n ) n ∈ N , and the labels of edges can be forgotten. For each λ i , the nextsymbol λ i +1 will be either v ( λ i ) or h ( λ i ). Since v ( λ i ) = h ( λ i ), the symbol λ i +1 uniquely determines the edge crossed by the corresponding geodesic.As Vincent Delecroix remarked to the author, it may be possible that the se-quence of labels ( λ n ) n ∈ N uniquely determines a cutting sequence ( λ n , e n ) even if h ( λ ) = v ( λ ) for some λ . On the other hand, this is not always the case, as shown inthe following example due to Pat Hooper. Suppose Λ = { , , ..., d } and let h = v be the permutation (1 2 3 · · · d ). The sequence of the labels of edges crossed byany geodesic on the surface is the periodic sequence ( h n (1)) n ∈ Z , and so the cuttingsequences are not completely determined by the labels of the squares. References [Arn02] Pierre Arnoux. Sturmian Sequences. In
Substitutions in dynamics, arithmetics and com-binatorics , volume 1794 of
Lecture Notes in Mathematics , chapter 6, pages 143–198.Springer-Verlag, Berlin, 2002. Edited by V. Berth´e, S. Ferenczi, C. Mauduit and A.Siegel.[Dav13] Diana Davis. Cutting sequences, regular polygons, and the Veech group.
Geom. Dedi-cata , 162:231–261, 2013.[Dav14] Diana Davis. Cutting sequences on translation surfaces.
New York J. Math. , 20:399–429,2014.[DPU15] D. Davis, I. Pasquinelli, and C. Ulcigrai. Cutting sequences on Bouw-M \ ”oller surfaces:an S-adic characterization. ArXiv e-prints , September 2015, 1509.03905.[FZ08] S´ebastien Ferenczi and Luca Q. Zamboni. Languages of k -interval exchange transforma-tions. Bull. Lond. Math. Soc. , 40(4):705–714, 2008.[GJ00] Eugene Gutkin and Chris Judge. Affine mappings of translation surfaces: geometry andarithmetic.
Duke Math. J. , 103(2):191–213, 2000.[Kea75] Michael Keane. Interval exchange transformations.
Math. Z. , 141:25–31, 1975.[KMS86] Steven Kerckhoff, Howard Masur, and John Smillie. Ergodicity of billiard flows andquadratic differentials.
Ann. of Math. (2) , 124(2):293–311, 1986.[MH38] Marston Morse and Gustav A. Hedlund. Symbolic Dynamics.
Amer. J. Math. ,60(4):815–866, 1938.[MT02] Howard Masur and Serge Tabachnikov. Rational billiards and flat structures. In
Hand-book of dynamical systems, Vol. 1A , pages 1015–1089. North-Holland, Amsterdam, 2002.[Sch04] Gabriela Schmith¨usen. An algorithm for finding the Veech group of an origami.
Exper-iment. Math. , 13(4):459–472, 2004.[Ser85] Caroline Series. The geometry of Markoff numbers.
Math. Intelligencer , 7(3):20–29,1985.[SU11] John Smillie and Corinna Ulcigrai. Beyond Sturmian sequences: coding linear trajecto-ries in the regular octagon.
Proc. Lond. Math. Soc. (3) , 102(2):291–340, 2011.[Vee89] W. A. Veech. Teichm¨uller curves in moduli space, Eisenstein series and an applicationto triangular billiards.
Invent. Math. , 97(3):553–583, 1989.[Wri15] Alex Wright. Translation surfaces and their orbit closures: an introduction for a broadaudience.
EMS Surv. Math. Sci. , 2(1):63–108, 2015.[WZ15] ShengJian Wu and YuMin Zhong. On cutting sequences of the L -shaped translationsurface tiled by three squares. Sci. China Math. , 58(6):1311–1326, 2015.[Zor06] Anton Zorich. Flat surfaces. In