TTiling billiards and Dynnikov’s helicoid
Olga Paris-Romaskevich
To Anatoly Stepin, who helped me do my first steps as a researcher.
Abstract
Here are two problems. First, understand the dynamics of a tiling billiard ina cyclic quadrilateral periodic tiling. Second, describe the topology of connectedcomponents of plane sections of a centrally symmetric subsurface S ⊂ T of genus3. In this note we show that these two problems are related via a helicoidal con-struction proposed recently by Ivan Dynnikov. The second problem is a particularcase of a classical question formulated by Sergei Novikov. The exploration of therelationship between a large class of tiling billiards (periodic locally foldable tilingbilliards) and Novikov’s problem in higher genus seems promising, as we show inthe end of this note. Bibliography : items; figures; MSC: Primary 37E35, Secondary 37J60;keywords : Novikov’s problem, tiling billiards, billiards, translation surfaces Tiling billiards are billiards in tilings. They were first introduced only several yearsago, in the works of Davis and her coauthors, see [7, 5, 8]. The definition of thebilliard flow is as follows. Each time a ray of light crosses an edge between two tiles, itrefracts through this edge. A new direction of the beam is obtained from the old one byreflection with respect to the crossed edge, following thus the Snell’s law of refractionwith coefficient −
1, see Figure 1.The goal is to understand the dynamics of such tiling billiards. What are typicaltrajectories? And atypical ones? The answers to these questions, and the dynamics ingeneral, depend strongly on the form of the underlying tiling.Figure 1: Four beams of light crossing a horizontal line under tiling billiard law.1 a r X i v : . [ m a t h . D S ] F e b o this date, our community has reached relative success in the understanding ofthe dynamics of three non-trivial tiling billiards. These are, trihexagonal tiling [8]; periodic triangle tilings [5], [18], [24]; and periodic cyclic quadrilateral tilings [12].In particular, the dynamics of a tiling billiard in a parallelogram tiling seems at themoment completely obscure.One of the reasons to be interested in tiling billiards is their connection to classicalobjects in mathematics. For example, the dynamics of triangle tiling billiards is equiv-alent to that of Arnoux-Rauzy family of interval exchange transformations and theirrel-deformations. The exceptional set of trajectories of these tilings is parametrized bya famous fractal object, the Rauzy gasket , see [3] for its definition.In this work we point out a new connection of triangle and cyclic quadrilateraltiling billiards to another classical subject which is a so-called Novikov’s problem.This problem studies the connected components of plane sections of triply periodicsurfaces. Of particular interest are chaotic components – curves such that there closurein the fundamental domain of the surface fills in the subsurface of genus at least 3.Novikov’s problem has deep connections to conductivity physics and is often presentedas the problem on semiclassical motion of an electron in a homogeneous magnetic field.We send our reader to [22] for the overview of the state of art on Novikov’s problemfrom the point of view of experimental physics. In this note we formulate and studythis problem in purely topological terms. Mathematically, first observations were doneby Zorich in [25], and a breakthrough via Morse theory has been done by Dynnikov, werefer especially to [11]. The generalizations of the problem related to quasi-periodicfunctions having more than 3 quasiperiods appear in [23].The connection between tiling billiard systems and Novikov’s problem has beenhinted to us by Dynnikov. His constuction gives a title to this work, and we make itexplicit.Once the connection between two subjects, tiling billiards and Novikov’s problem,is established, we show how the ideas from topology and Morse theory that were de-velopped for treating the Novikov’s problem, apply to tiling billiard dynamics. Weformulate, at the very end of this work, Conjecture 5.1 on the behavior of tiling bil-liards in a large class of tilings that includes triangle and cyclic quadrilateral tilings - locally foldable tilings . This conjecture is based on two strongly non-trivial results.First, Dynnikov’s result on the generic behaviour of plane sections of 3-periodic sur-faces that he obtained at the end of the last century in [11]. And second, a recent resultby Kenyon, Lam, Ramassamy and Russkikh [19] describing the space of parameters oflocally foldable tilings, in the setting of the dimer model.The general intention of this article is to express hope for a new approach of theNovikov’s problem through tiling billiard dynamics and, in particular, renormalizationfor such dynamics. We refer our reader to [24] and [12] for introduction to renormal-ization in tiling billiards. Recently we have shown, in collaboration with Dynnikov,Hubert, Mercat and Skripchenko, that the measure of chaotic regimes in the Novikov’sproblem with central symmetry, in genus 3, is equal to 0. This question has been openfor 40 years and we answer it using the renormalization of cyclic quadrilateral tilingbillards. We hope that the connection with tiling billiards will permit new discoveriesfor Novikov’s problem in higher genus as well.2his note is structured in a following way. In Section 2, we remind the foldingprocedure for tiling billiards and define locally foldable tilings, and, in particular, tri-angle and cyclic quadrilateral periodic tilings. The billiards in these two last classes oftilings are the main dynamical systems discussed in this note. The Section 2 introducesthe folding map and the so-called parallel foliations are obtained as preimages underfolding of standard foliations by parallel lines. In Section 3 triply periodic surfacescalled Dynnikov’s helicoids are constructed, correponding to tiling billiards. In Sec-tion 4 we remind the statement of Novikov’s problem (paragraph 4.1) and make explicitthe connection between tiling billiards and this problem, using Dynnikov’s helicoids.Then, we interpret classical results on Novikov’s problem in terms of such tiling bil-liards (paragraph 4.2) and advance in the proof of the so-called Tree Conjecture forcyclic quadrilateral tilings (paragraph 4.3). Sections 3 and 4 concern only triangle andcyclic quadrilateral tilings. In Section 5, we discuss open questions and perspectivesfor general locally foldable tilings.We excuse ourselves for some familiarity with regularity. The goal of this articleis to share the ideas on a conceptual level. The question of regularity of the surfacesin Novikov’s problem is although extremely important, and should be addressed if onewants to obtain precise statements. It will be done in the upcoming work [12], at leastfor cyclic quadrilateral tilings.
Any triangle (or quadrilateral) P tiles a plane periodically as follows. A fundamentaldomain D m of a tiling is obtained by gluing P with its centrally symmetric copy, withrespect to a middlepoint m of its side. We call a corresponding P -tiling P a (periodic)triangle (quadrilateral) tiling . Such tiling is 2 -colorable in a way that neighbouringtiles have different colors, as a chess-board. We denote P a tiling and the set of its tiles.A tiling of a plane by polygons is locally foldable if it is 2-colorable, and thesum of angles in any vertex v of the same-colored tiles containing v is equal to π .Triangle tilings are locally foldable, and quadrilateral tilings are locally foldable onlyif the quadrilateral is cyclic (inscribed in a circle). Tiling billard trajectories in locallyfoldable tilings share fundamental properties that we state in the points 2 . and 3 . ofTheorem 2.1 below. Throughout this work, we concentrate on the case of P -tilings bytriangles and cyclic quadrilaterals. In Section 5 we discuss the general case. I like to fold my magic carpet, after use, in such a way as to superimpose one part ofthe pattern upon another.
Vladimir Nabokov,
Speak, Memory
Opening up a polygonal billiard table in order to understand the trajectories is nowa habit for any mathematician : while a table is unfolded in order to produce a, poten-tially, self-overlapping and layered tiling of the plane, a trajectory is unfolded into astraight line. This idea (called Katok-Zemlyakov construction) provides, in the case of3ational tables, a connection of billiard dynamics with translation flows. For tiling billiards, the billiard table is already a tiled plane. We fold two neigh-bouring tiles along the crease like the wings of an asymmetric butterfly. The segmentsof a tiling billiard trajectory in these two tiles fold into segments on the same line. Itis easy to show that for locally foldable tilings, such a folding map is defined globally,and not only along a path in a tiling. The entire plane may be folded and any tiling bil-liard trajectory folds into a line inside this folding, see [24] for more details and precisestatements. In the following, we use this folding map, unique up to isometry.For triangle and cyclic quadrilateral tilings, the image of a folded plane is partic-ularly simple to understand. The plane folds inside a disk, and all of the vertices ofa tiling fold onto its boundary - a circle. The preimage of this circle under the fold-ing is the union of all circumcircles of tiles. This beautiful observation first appearedin the work [5] by Baird-Smith, Davis, Fromm and Iyer. Their work contains manyillustrations, as well as a pattern to cut out and experience the folding manually.A powerful, and elementary, consequence of the existence of global folding is
Theorem 2.1 ([5], [18], [24]) . The following holds for the trajectories of tiling billiardsin periodic triangle and cyclic quadrilateral tilings.1. The oriented distance τ ( γ , P ) between an (oriented) segment of a trajectory γ in atile P and the circumcenter of this tile, is constant along γ , that is, τ ( γ , P ) = τ ( γ ) ;2. every trajectory intersects any tile in at most one segment;3. any bounded trajectory is periodic and stable under small perturbations (of atile or initial condition): perturbed trajectory passes by the same tiles. The authors of [5] use the folding in order to reduce the dynamics of the triangletiling billiard to that of a family of 3-interval exchange transformations (with flips!) ona circle. For cyclic quadrilaterals, similarly, one gets 4-IETs, see paragraph 3.4 here. InSection 3 of our work [24], Theorem 2.1 is generalized for all locally foldable tilings.
Take any periodic trajectory γ of a billiard in a tiling, not even necessarily locallyfoldable. A close enough to γ trajectory γ (cid:48) , launched in the same direction as γ , isnecessarily periodic and disjoint from γ . The cylinder between γ and γ (cid:48) is foliated byparallel tiling billiard trajectories.For triangle and cyclic quadrilateral tiling billiards this idea can be pushed muchfurther. First, as follows from Theorem 2.1, «in the same direction» can be omitted fortriangle and cyclic quadrilateral tilings. And second, parallel trajectories foliate the fullplane, as we noticed and explored in [24]. For any trajectory γ , there exists a so-called parallel foliation of the entire tiled plane (singular only in vertices of the tiling) suchthat its non-singular leaves are tiling billiard trajectories (one of which is γ ) and that For tiling billiards, the connection with translation flows is not straightforward, even though it may begiven, at least for locally foldable tilings.
4n restriction to any tile, it is a foliation by parallel segments. We invite our reader todiscover a video work
Refraction tilings by O. David on parallel foliations. Remark 2.2.
It is interesting to compare the parallel foliation construction for tilingbilliards with the straight skeleton method for polygons [1] discovered in 1995. Thismethod was used by Demaines, father and son, and Lubiw, see [10], in order to solvea following fold-and-cut problem. Fix a polygonal motive on a piece of paper. Can thepaper be folded in such a way that a polygonal motive may be cut out by exactly onescissor cut ? (and the answer is yes!)
Parallel foliations are a handy tool to study the dynamics of locally foldable tilingbilliards. Indeed, to one trajectory is associated an entire family of trajectories in thesame parallel foliation. The singular leaves of such a foliation uniquely define thesymbolic dynamics of all other leaves. We have introduced parallel foliations in [24]in order to prove the following Tree Conjecture (a theorem since), formulated in [5].
Theorem 2.3 ([24]) . For any periodic trajectory γ of a triangle tiling billiard, thedomain Ω γ bounded by it doesn’t contain any full tile of the tiling. In other words, allthe vertices and edges of the tiling contained in Ω γ form a graph which is a tree. In paragraph 4.3, we advance in the proof of the Tree Conjecture for cyclic quadri-lateral tiling billiards. For this, we use a new idea that connects such tiling billiardswith the topology of sections of periodic surfaces. This idea is a heart of this articleand deserves a section for itself.
Here we construct a helicoidal surface such that the connected components of its hori-zontal sections coincide with tiling billiard trajectories of the same parameter τ ( γ ) = τ ,see Theorem 2.1. This surface will be triply periodic and, via projection, a compactsubsurface of the 3-torus. The study of trajectories will be thus reduced to the studyof plane sections of surfaces in T which is a classical problem discussed in Section 4.We speculate in [24] on the existence of such a link between this problem and tilingbilliards. Recently, Dynnikov made this connection precise, and we make his brilliantidea explicit. For a triangle (or cyclic quadrilateral) tiling P , we fix the following notations that arerespected throughout the article, in particular on Figure 2. Triangle tiling.
The sides of the tiles are a , b and c , a clockwise tour of a tile reads abc . The vertices opposite to the sides a , b , c are A , B , C and their angles are α , β and γ .Define vectors a : = BC , b : = CA , c : = AB . Then a + b + c = α + β + γ = π . Cyclic quadrilateral tiling.
The sides of the tiles are a , b , c and d , a clockwise tourof a tile reads abcd . The vertices are A , B , C , D and A = d ∩ a , B = a ∩ b etc. The angles The video is accessible on the Youtube channel
Dragonazible, or via https://youtu.be/t1r1cO1V35I.
5n these vertices are α , β , γ and δ . Define vectors a : = AB , b : = BC , c : = CD , d : = DA .Cyclicity is equivalent to the relations α + γ = β + δ = π . Fix a tile P ∈ P . There exists a unique folding, in sense of paragraph 2.1 such that thetile P is fixed. Then, to any tiling billiard trajectory γ , we associate two parameters.First, we define the energy τ ( γ ) via Theorem 2.1. It is positive if the trajectory turnscounterclockwise around the circumcenter and negative otherwise. When τ ( γ ) =
0, thedirection of the trajectory points in (or out of) the circumcenter. The set of energyvalues is bounded and symmetric. Up to a rescaling of the tiling, we suppose that τ ∈ [ − π , π ] .The second parameter is the angle θ ( γ , P ) that makes an (oriented) segment of atrajectory in a tile P with some fixed direction. This parameter depends on a tile thatthe trajectory γ crosses. We denote by θ ( γ ) : = θ ( γ , P ) . If some trajectory γ doesn’tcross a tile P , the value θ ( γ , P ) can still be defined and is done via the following Lemma 3.1.
For any tile P in triangle (or cyclic quadrilateral) tiling, there exists aunique function f P ( θ ) : = f ( P , θ ) : P × [ , π ) → [ , π ) such that f ( P , θ ) ≡ θ and thatthe following holds.1. For two tiles P and P of same (different) color,f P ( θ ) ∓ f P ( θ ) = ϕ ( v ) , v : = P − P . Here ϕ is a function of the vector v that connects the barycenters of the tiles.
2. For any trajectory γ and any two tiles P and P that it crosses, θ ( γ , P ) ∓ θ ( γ , P ) = ϕ ( v ) . Hence θ ( γ , P ) is globally and correctly defined for all γ and all P ∈ P , even if γ ∩ P = /0 .3. The function f is defined via Figure 2 in the case when the fixed direction is thatof AB. In general, it is sufficient to define its values on all the tiles neighbouringto P and then to continue by quasiperiodicity on all P .Proof. This follows obviously from the existence of folding. The images of two tilesof the same color P , P map under folding differ by a circle rotation by ϕ ( P − P ) .The function f P ( θ ) is described also in Table 1 of [5] for triangle tilings. Although,there it is only defined on «half» of the tiles. On Figure 2 we picture the values ofthe function f on the tiles that one can access in one or two steps from P . A Lemmaanalogous to Lemma 3.1 can be proven for any periodic locally foldable tiling. Remark 3.2.
If a fixed direction is defined by some angle θ then, on Figure 2, oneshould add θ to all values of ϕ ( P − P ) with P- grey and subtract it when P-white. In other words, this vector depends only on the relative positions of tiles. This vector belongs to theperiod lattice for tiles of the same color. f P ( θ ) ± f P ( θ ) in the vicinity of P , for triangle andcyclic quadrilateral tilings. The values are calculated using the notations of paragraph3.1. In both cases, a fixed direction is chosen as that of the segment AB in P . In this paragraph we construct a one-parametric family { (cid:99) S τ P } τ ∈ [ − π , π ] of piecewise smoothsurfaces in R , associated to any triangle (cyclic quadrilateral) tiling P . All the trajec-tories with the same energy τ appear as horizontal sections of (cid:99) S τ P .Fix some parameters τ ∈ [ − π , π ] and θ ∈ [ , π ) . Let Γ τ , θ be the set of all tilingbilliard trajectories γ such that τ ( γ ) = τ and θ ( γ , P ) = θ . We identify the set Γ τ , θ andthe geometric union of the trajectories ∪ γ ∈ Γ τ , θ γ . Potentially (and generically, as shownin [18], [24] and [12]), the set Γ τ , θ consists of more than one trajectory. This set isobtained as the union of curves which fold into the same chord defined by parameters τ and θ . Example 3.3.
Fix some θ ∈ [ , π ) . Then the sets { Γ τ , θ } τ ∈ [ − , ] foliate the tiled planeand form a parallel foliation corresponding to any trajectory γ with θ ( γ , P ) = θ . Let us now fix τ . Then the sets Γ τ , θ overlap, even inside one tile. Similarly tothe case of the geodesic flow, we lift them up in another dimension, in order to avoidintersection.Consider the euclidian space R = { ( X , Θ ) } as a product of a tiled plane R with acoordinate X on it, and of an orthogonal line with a coordinate Θ . We now define a set (cid:99) S τ P as a union of its horizontal sections: (cid:99) S τ P : = ∪ θ ∈ [ , π ) Γ τ , θ . It is obviously a piecewisesmooth surface.The surface (cid:99) S τ P is built from many gradually turning «stairs» of trajectories with thesame energy parameter. Any tiling billiard trajectory γ on the tiling P is a connectedcomponent of a horizontal section of the helicoid, its height is defined by the angleparameter. We call such a surface (cid:99) S τ P (depending strongly on the underlying tiling) Dynnikov’s helicoid of energy τ . It has 3 periods that do not depend on τ , as showsthe following 7igure 3: The compact ruled surface S τ P obtained as an intersection of Dynnikov’shelicoid with a torus obtained by gluing parallel faces of the prism D × [ , π ) . Lemma 3.4.
Let P be a triangle or cyclic quadrilateral tiling, and (cid:99) S τ P a correspondingDynnikov’s helicoid of some energy τ ∈ [ − π , π ] . Then, (cid:99) S τ P is -periodic with periodsV : = V ( P ) , V : = V ( P ) and V = ( , , π ) given by1. V ( P ) = ( − c , γ ) , V ( P ) = ( − a , α ) if P is a triangle tiling,2. V ( P ) = ( a + b , δ ) , V ( P ) = ( b + c , α ) if P is a cyclic quadrilateral tiling.The notations here are consistent with those from paragraph 3.1.Proof. Periodicity in vertical direction is obvious since Γ τ , θ = Γ τ , θ + π . The rest followsfrom Lemma 3.1. Indeed, for two tiles P , P of the same color, the difference of angleparameters is constant and equal to ϕ ( v ) with v = P − P . One concludes that a point ( X , Θ ) ∈ (cid:99) S τ P belongs to a surface if and only if a shifted point ( X (cid:48) , Θ (cid:48) ) : = ( X + v , Θ − ϕ ( v )) ∈ (cid:99) S τ P does. By continuity, the argument follows for differently colored tiles.The lattice of periods is generated by the vectors v : = −→ BA = − c and v : = −→ CB = − a for triangle tilings and by the vectors v : = −→ AC = a + b and v : = −→ BD = b + c for cyclicquadrilateral tilings. The values of ϕ are given on Figure 2. S τ P and link to interval exchange maps We now describe some properties of Dynnikov’s helicoid. Consider a fundamentaldomain D m of a P -tiling, as in Section 2 and on Figure 4. Fix a horizontal directionas the direction needed for the definition of the angle parameter as that of the edgecontaining m .Lemma 3.4 implies that the intersection with the prism D m × [ , π ) ∩ (cid:99) S τ P is a fun-damental domain of the surface (cid:99) S τ P . Under identification of the borders under shifts8igure 4: Fundamental domain D m of a triangle tiling and two trajectories passingthrough it, γ and γ (cid:48) . Each of the trajectories is tangent to both circles centered at cir-cumcenter of radius τ ( γ ) = τ ( γ (cid:48) ) = τ . Moreover, θ ( γ ) + θ ( γ (cid:48) ) = π . V j , j = , ,
3, such a prism becomes a 3-torus T P . Let π : R → T P be a correspondingprojection. Denote by S : = S τ P = π (cid:16)(cid:99) S τ P (cid:17) a compact surface, represented on Figure 3. Lemma 3.5.
Let τ ∈ [ − π , π ] , and P be a triangle (cyclic quadrilateral) tiling. For acorresponding Dynnikov’s helicoid (cid:98) S : = (cid:99) S τ P and S : = π ( (cid:98) S ) , the following holds:1. the surface S is centrally symmetric with respect to the point M : = ( m , π ) . Thequotient S M under the central symmetry with respect to M, is a non-orientablesurface; moreover, this point M belongs to S if and only if τ = ;2. if τ = , the surface S has an additional symmetry under the map s : Θ (cid:55)→ Θ + π .The quotient S / s under this symmetry is homeomorphic to the projective plane P ( R ) . Moreover, for tiles containing its circumcenter, the foliation on the pro-jective plane induced by a horizontal foliation on S is a foliation with one (or )-prong singularity and three (or four) -prong singularities;3. if P contains its circumcenter and τ = , the genus of S is equal to . For P atriangle, S has two double saddles, exchanged by s . If P is a cyclic quadrilateral,the surface S has four simple saddles, exchanged in pairs via s . If P doesn’tcontain its circumcenter, the genus of S is equal to .Proof. All of this is direct. The first statement follows from the symmetry of D m .If there exists a trajectory γ crossing D m × { θ } , there exists a trajectory γ (cid:48) crossing D m × { π − θ } , see Figure 4. If τ =
0, the trajectory passing by m and orthogonal tothe edge that contains it, belongs to S , and γ coincides with γ (cid:48) above (and has oppositeorientation). The second statement follows from the existence of the additional sym-metry and the calculation of Euler characteristic. In this particular case, the interiorand exterior of S P in T P are isometric.Suppose now that P contains its circumcenter and τ =
0. We calculate χ ( S ) as asum of indices of its singular points with respect to the height function Θ . If P is a for triangles it is equivalent to the acuteness D m are identified under lattice action. The surface S has then two monkey saddles (index −
2) corresponding to the angle parameters θ and θ + π , where the trajectories enter (or get out from) the vertex. This gives χ = − g =
3. Analogous calculation may be done for «acute» cyclic quadrilaterals : D m hastwo vertices, modulo the action of the lattice. They give then 4 simple saddles on layers θ , θ , θ + π , θ + π , each of index −
1. Then, once again, g =
3. For the obtuse case,the same calculation gives g =
1. This proves the third point.
Remark 3.6.
It is not surprising that genus may fall drastically when the parameterschange continuously since the helicoid is parametrically defined by θ and τ . A goodexercise is to understand the change of genus of S when τ changes. The surfaces (cid:98) S and S are equipped with natural oriented foliations by tiling billiardtrajectories. The intersection of S with the border of prism D m × [ , π ) consists of acircle, and the first return map of the tiling billiard flow on this circle is an intervalexchange transformation T = T ( P , τ ) with 6 or, in case of quadrilaterals, 8 intervalsof continuity. If one passes to the quotient S M , such first return map is reduced to aninterval exchange transformation F ( P , τ ) of 3 or 4 intervals of continuity with flips, and T = F . The study of tiling billiard dynamics in triangle and cyclic quadrilateral tilingsis then reduced to the study of parametric families of interval exchange transformationswith flips. This is the important leitmotiv of the works [5, 18, 24] for triangles and ofthe work [12] for quadrilaterals. The helicoid construction from Section 3 is elementary but crucial since it connectsthe dynamics of tiling billiards with a classical topology problem formulated in 1982by Novikov. It concerns the level sets of quasiperiodic functions on the plane with 3quasi-periods and has important motivations coming from physics of metal conductiv-ity. Nowadays, the interest to this problem is vivid in both mathematics and physics.Novikov’s problem is a field in itself and we do not aim to give an overview nor a bib-liography of this rich subject. Our goal is to point out a new connection – the one withtiling billiard systems that, hopefully, can shed some light on the problem in itself.
Consider a piecewise smooth function f : T → R , T = R / Z . Without loss of gener-ality we suppose that f takes values in the interval [ − π , π ] . Let M τ = f − ( τ ) be its levelsurface and (cid:98) M = π − ( M ) be the Z -covering in R . Here π : R → T is a standardprojection. Novikov’s problem.
Fix a covector H = ( H , H , H ) ∈ P ( R ) . Study the behaviorof connected components of plane sections of (cid:98) M by a family of parallel planes H x + H x + H x = const.The corresponding parallel plane sections define an orientable foliation F on M .We are interested in the closures of its leaves.10or simplicity we suppose that the covector H is totally irrational. As Dynnikovshowed in [11], three qualitative behaviors are possible: trivial, integrable and chaotic. Trivial behavior means that all components of all H -sections are compact. Integrablebehavior means that all regular non-closed components are confined in bands of theplane, i.e. have an asymtptotic direction. Integrable behavior corresponds to the de-composition of M into cylinders of closed trajectories and tori (possibly, with holes)on which F winds in a way that it is topologically equivalent to irrational rotation.Finally, chaotic behavior means that the closure of some leaf of F coincides with acomponent of M of genus at least 3. One of the main results in [11] is that the chaoticbehavior occurs in a very rare number of cases, see Theorems 1 and 3 there. Theorem 4.1 (Dynnikov, [11]) . Fix a piecewise smooth and generic function f anda vector H. Then there exist two values τ ( H ) , τ ( H ) , τ ≤ τ such that for all τ / ∈ [ τ ( H ) , τ ( H )] the behavior of corresponding sections of M τ is trivial, and for τ ∈ [ τ ( H ) , τ ( H )] it is integrable. In the case when τ ( H ) = τ ( H ) , the behavior may bechaotic. Moreover, the chaotic behavior is rare in the following sense : the set O ofvectors H corresponding to non-chaotic behavior is open and dense in P ( R ) . Question 1.
The question of whether the set O is of full Lebesgue measure is open,even for M of genus . Remark 4.2.
The space of couples function-covector { ( f , H ) } has infinite dimensionalthough the qualitative behavior of sections depends only on the finite number of pa-rameters. Indeed, one considers an exact -form induced on M by a linear form α = dHon R . Via Hodge theorem, take a unique harmonic form ω on M such that ω ∈ [ α ] .Then ω defines a cohomologous, and even cobordant foliation to F which has the sameglobal invariants as F itself. In other words, one can straighten out the foliation F and preserve the class of qualitative behavior (trivial, integrable or chaotic). The cor-responding surface has a flat metric. This shows the relationship of Novikov’s problemwith the dynamics of families of interval exchange transformations.Moreover, the word generic in Theorem 4.1 has to be precised. We refer our readerto the original article [11]. Some details are discussed at the end of this Section. Question 2.
A much less stronger question than Question 1 is open - prove that in theset of pairs (surface, vector) the set of chaotic couples is of measure . The most strongest form of such type of questions is a following
Conjecture 4.1 (Novikov-Maltsev, 2003) . For a fixed surface M, the Hausdorff di-mension of the set of covectors H admitting chaotic sections is smaller than . Our goal here is to include the study of tiling billiards in triangle and cyclic quadrilat-eral tilings into the setting of Novikov’s problem.Lemma 3.4 states that a surface (cid:99) S τ P is 3-periodic, with the vectors V , V , V definingthe base of the corresponding lattice of symmetries. Therefore, there exists a unique11inear map A P ∈ SL ( R ) such that A P ( V j ) = E j , with E j forming the standard or-thonormal basis in R , j = , ,
3. Note that V = π · E . Denote (cid:99) M τ P : = A P (cid:99) S τ P therectified surface. Then, M τ P : = π ( (cid:99) M τ P ) is a subsurface of a standard torus. Traectoriesof a billiard are the connected components of horizontal sections of Dynnikov’s heli-coid. Under the linear map, they map to the connected components of intersections (cid:99) M τ P ∩ A P { Θ = θ } . Once a helicoid is constructed, two points of view differ only by alinear map! Remark 4.3.
The map A P as well as the direction of the co-vector H defining theplanes A P { Θ = θ } doesn’t depend on τ since the vectors V j do not depend on it. Question 3.
How large is a class of surfaces described (in terms of Remark 4.2) byDynnikov’s helicoids for triangle and cyclic quadrilateral tilings?
The first non-trivial case of Novikov’s problem (when chaotic behavior is possible)occurs in genus g ( M ) =
3. By Lemma 3.5, the maximal genus of Dynnikov’s helicoidsis equal to 3 and such helicoids are always centrally symmetric. We think that this isthe only obstruction and that the answer to the Question 3 is : any centrally symmetricsurface of genus
3. A careful dimension count should be done, see Remark 4.2.Let us now remind some results on the dynamics of considered tiling billiards. Thefollowing has been conjectured in [5], first proven in [18] and, finally, a simpler proofwas found in [24] via renormalization. As we have recently discovered, the analogousproof, even if in a different setting, has already been provided in 1989 by Meester andNowicki in [21]. Theorem 4.4 ([18, 24, 4]) . For a tiling billiard in a P-tiling defined by a triangle P,the following holds :1. For almost any P, all trajectories are either periodic or linearly escaping.2. If a trajectory γ escapes in a non-linear way then, necessarily, τ ( γ ) = (it passesby circumcenters of all crossed tiles) and P ∈ R . Here R is a fractal set de-fined by an explicit continued fraction algorithm. This set has zero measure and dim H R ∈ ( , ) . This set R is the Rauzy gasket . We refer to [3] for the classic definition of theRauzy gasket. Many other definitions have been given throughout the last fourtyyears, related to the circle percolation (Meester-Nowicki [21]), dynamics of Arnoux-Rauzy family of interval exchange transformations (Arnoux-Rauzy [2]), systems ofisometries (Dynnikov-Skripchenko [14]), Novikov’s sections of some polyhedral ob-ject (Dynnikov-DeLeo [9]), and, as shown here, in relation to the dynamics of non-linearly escaping trajectories of tiling billiards (Davis et al. [5], Hubert and ourselves Meester and Nowicki consider a percolation model on the circle which is exactly the circumcircle ap-pearing via folding. Their model is defined by drawing a chord in a circle and coloring all vertices in theset { α n + β m , m , n ∈ Z } ⊂ S on the left of the chord in one color, and others in another color. The corre-sponding coloring of the lattice Z with coordinates ( m , n ) has open one-colored clusters. They correspondto escaping trajectories of triangle tiling billiards that fold into the initial chord. Remark 4.5.
The calculation of the Hausdorff dimension of the Rauzy gasket is highlynon-trivial. The first estimate dim H R < was obtained in [4]. Nowadays more refinednorms exist and dim H is confined into the interval ( . , . ) . The lower bound wasobtained by Gutiérrez-Romo and Matheus in [17], the upper bound is obtained bycombining the arguments of Gamburd-Magee-Ronan [16] with estimates by Baragar[6], as was recently explained in [15]. Recently, our French-Russian team managed to prove the following result, analo-gous to Theorem 4.4 result for cyclic quadrilateral tilings.
Theorem 4.6 ([12]) . For a tiling billiard in a P-tiling defined by a cyclic quadrilateralP, the following holds :1. For almost any P, all trajectories are either periodic or linearly escaping.2. If a trajectory γ escapes in a non-linear way then, necessarily, τ ( γ ) = andP ∈ N . Here the set N is a fractal set defined by an explicit continued fractionalgorithm, has zero measure and dim H N < . The first difficulty in proving this Theorem was to find a renormalization processin order to define the algorithm that constructs N via a continued fraction algorithm.Once this was done, the main technical difficulty consisted in proving the ergodic prop-erties of such an algorithm. The first part is combinatorial and generalises the methodsin [24], the second part is based on the thermodynamic formalism, elaborated recentlyby Fougeron [15].Modulo the regularity details described in Question 3 that have to be figured out,Theorem 4.6 solves an open case of Novikov’s problem since it characterizes chaoticdirections for symmetric genus 3 surfaces. In relation to this interpretation, the set N of cyclic quadrilaterals that may exhibit non-linear escaping trajectories is called the Novikov’s gasket.
The renormalization methods proposed in [24, 12] prove the first points of bothTheorems 4.4 and 4.6. Although one can see that these first points follow, naively ,from Dynnikov’s result of 1999, namely Theorem 4.1 here. Indeed, since for a fixeddirection H , the chaotic behavior can only happen for one energy parameter, the centralsymmetry of M τ implies that this parameter is exactly τ = f corresponding to some Dynnikov’s helicoidfor triangle tilings. Then f is not generic in Dynnikov’s sense : its level surfaces havedouble saddle points and the work [11] works with surfaces admitting only simplesaddles. Although, the explicit calculations of chaotic sections were done for this casein [24] and in [9], and they finalise the proof of Theorem 4.4.For cyclic quadrilaterals, Dynnikov’s results do apply since the surfaces are genericenough and the corresponding saddles are simple. The point 1. of Theorem 4.6 is then13 direct consequence of Theorem 4.1. We wonder if the genericity assumptions ofTheorem 4.1 in [11] could be weakened in order to apply directly to surfaces withsaddles of higher multiplicity. Remark 4.7.
Triangle tilings can be seen as degenerations of cyclic quadrilateraltilings with two vertices of a quadrilateral approaching by following the arc of thecircumcircle of a tile. The corresponding surfaces are degenerations of a more generalcase : saddle points collide in a double (monkey) saddle.
In this paragraph, we advance towards the understanding of the symbolic dynamics ofcyclic quadrilateral tilings, analogous to Theorem 2.3.
Conjecture 4.2 (Tree Conjecture for cyclic quadrilateral tiling billiards) . Any periodictrajectory γ in cyclic quadrilateral tiling doesn’t contour tiles, i.e. the domain Ω γ bounded by it doesn’t contain a full tile. The symbolic behavior of any periodic trajectory γ is defined by the behavior ofsingular trajectories in its parallel foliation inside Ω γ . Using this idea, we have shownin [24] that for any locally foldable tiling, the Tree Conjecture is equivalent to the fol-lowing Bounded Flower Conjecture dealing with only singular trajectories (or petals ). Bounded Flower Conjecture.
Any singular periodic trajectory γ passing by a ver-tex v of a tiling satisfies the two following properties : first, it intersects two neigboringtiles P and P e , P ∩ P e = e ; second, e ∈ Ω γ .Let us include a petal γ in its parallel foliation. If there is another petal γ (cid:48) passing bythe same vertex v , then we can prove that the Bounded Flower Conjecture for γ holds.If it doesn’t, it would mean that either Ω γ ⊂ Ω γ (cid:48) or Ω γ (cid:48) ⊂ Ω γ , and two trajectories haveopposite orientations. Then there exist two periodic trajectories of the same energy τ inthe parallel foliation turning in different senses. It means that a corresponding helicoid (cid:99) S τ P has a section with two connected components, one of electron type, and one of holetype in the terminology of [11]. This and the connectedness of (cid:99) S τ P would imply that thesurface S τ P has genus at least 4 which brings a contradiction with Lemma 3.5.Unfortunately, we were not yet able to eliminate the sitation when γ is an only sin-gular trajectory passing by v in its parallel foliation, giving the obstructions to BoundedFlower Conjecture. It would mean that either γ doesn’t pass by P and P e , or it does butwith e / ∈ Ω γ . In the case of triangle tilings, the two cases were eliminated by usingthe additional symmetries which are not anymore present for quadrilaterals. The sym-bolic dynamics for quadrilateral tiling billiards is more complicated and still needs tobe understood in more detail. The ideas of Section 2 apply to any locally foldable tiling: parallel foliations exist,and all bounded trajectories are periodic and stable. Moreover, the quasiperiodicityobserved in Lemma 3.1 is present in any locally foldable periodic tiling.14igure 5: Bipartitie graphs on the torus T defining the combinatorics of triangle andquadrilateral tilings. Both graphs have 2 vertices, the triangle graph has 3 edges, thecyclic quadrilateral graph has 4 edges. The torus is given as regluing the oppositeparallel sides of the fundamental domain of the tiling.In this Section we show why the helicoid of Section 3, constructed there for trian-gle and cyclic quadrilateral tilings, can be constructed for many other locally foldablepolygonal periodic tilings. Remark 5.1 (Combinatorial data of a locally foldable periodic tiling) . Any locally fold-able polygonal periodic tiling defines a bipartite graph G on the -torus T . Indeed,we consider a dual graph (a graph of faces of the tiling) : two tiles are connected ifand only if they have a common edge in the tiling, see Figure 5 for two examples. Byperiodicity, this graph factors to the graph G on the torus. Since the locally foldabletiling is -colorable, G is bipartite. Of course, one such graph G defines a family ofcorresponding locally foldable tilings. We say that these tilings have the same combi-natorics. The apparent difficulty in the realization of a helicoidal construction for a generallocally foldable periodic tiling is that it is not clear what should be the energy parameter τ . In the point 1. of Theorem 2.1 τ was defined as a distance to a circumcenter of atriangle or cyclic quadrilateral. Here we point out an approach that gives a substitute tothis circumcenter in the general case. For this, we use recent results obtained on locallyfoldable tilings in relationship to the study of dimers.It is non-trivial to describe the set of parameters of locally foldable periodic polyg-onal tilings of fixed combinatorics G . This can been done by following recent works ondimers, in particular that by Kenyon, Lam, Ramassamy and Russkikh in [19]. There theauthors show that there is a bijection beween such tilings and liquid phase dimer mod-els . This bijection uses the beautiful connection of the dimer model with the complexalgebraic curves and their amoebas established by Kenyon, Okounkov and Sheffield. Ithappens that the studied curves are of a very special type, namely Harnack curves, werefer to [20] for more details.From all this important theory we use only the fact that typically the locally foldabletilings fold into bounded domains. In this case, take any tile P and all of its copies gP in the tiling, here g is an element of the lattice of isometries of the tiling. Then,the boundedness of the folding implies that for all g , the images of P and g ( P ) underfolding differ by a rotation with some center C . Moreover, the center C can’t depend This corresponds to the spectral curve only having simple zeroes. Indeed, the orbit of a discrete subgroup H of the affine group is bounded only if H is a subgroup of P in order for the folding to be bounded. This center C is a point with respect towhich we define the energy τ ! Once this step is done, the helicoid construction ofSection 3 is repeated word by word. Question 4.
Suppose that the helicoid S τ P exists for a locally foldable tiling P . Whatis its genus as a function of P and τ ? Question 5.
What families T P of interval exchange transformations arise as first-return maps on some well-chosen transversals ? We hope to answer these questions in future work, in order to prove the following
Conjecture 5.1.
Fix the combinatorics of a periodic -colored tiling of a plane, definedvia a bipartite graph G on the -torus. Suppose that a tiling with such combinatoricsfolds into a bounded domain. Then, the non-linear escape of trajectories on such tilingis only possible if the trajectories pass by the point C. Otherwise, the trajectories eitherescape linearly or are periodic. Conjecture 5.1 follows naively from Theorem 4.1 and helicoidal construction. Wedo not announce it as a result since the regularity details have to be thouroughlychecked, as discussed in paragraph 4.2. Indeed, locally foldable tilings permit mut-liplicity in saddles, although the arguments in [11] suppose Morse property. Never-theless, we believe that these complications are avoidable. Moreover, as follows fromdimer model theory, this Conjecture would apply to an open set of parameters of locallyfoldable tilings, maybe even of full measure.A following much stronger conjecture is a reformulation of Question 2.
Conjecture 5.2.
The set of parameters of locally foldable tilings of fixed combinatoricsG admitting non-linearly escaping trajectories, has measure zero.
Question 6.
Is such set an invariant gasket of some continued fraction algorithm?
We find very exciting a possibility to construct multi-dimensional fractal objectscorresponding to every bipartie graph on the torus. It could make quite a collection! Forthe graphs, corresponding to triangle and quadrilateral periodic tilings, these objectsare, respectively, the Rauzy and the Novikov gaskets.
ACKNOWLEDGMENTS
I am grateful to Ivan Dynnikov for his beautiful idea of a helicoid that he shared in ashort on-line call during the pandemic, Section 3 is entirely based on it. I am grateful toPascal Hubert and Bruno Sevennec for fruitful discussions on the subject and commentson the preliminary versions of this text. I am thankful to Dima Chelkak for introducingme to the dimer model and Benoît Laslier for answering my questions on it and advice.I am obliged to Théo Marty for the Figure 3 he drew in Inkscape in one evening and toPaul Mercat for his 3D prints of Dynnikov’s helicoids. I am also thankful to my newhome,
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O. Paris-Romaskevich A IX M ARSEILLE U NIV , CNRS, C
ENTRALE M ARSEILLE , I2M, M
ARSEILLE ,F RANCE
E-mail address
O. Paris-Romaskevich: [email protected]@math.cnrs.fr