Internal-wave billiards in trapezoids and similar tables
IInternal-wave billiards in trapezoids and similar tables
Marco Lenci ∗† , Claudio Bonanno ‡ , Giampaolo Cristadoro § February 2021
Abstract
We call internal-wave billiard the dynamical system of a point particle thatmoves freely inside a planar domain (the table ) and is reflected by its boundaryaccording to this rule: reflections are standard Fresnel reflections but with thepretense that the boundary at any collision point is either horizontal or vertical(relative to a predetermined direction representing gravity). These systemsare point particle approximations for the motion of internal gravity waves inclosed containers, hence the name. For a class of tables similar to rectangulartrapezoids, but with the slanted leg replaced by a general curve with downwardconcavity, we prove that the dynamics has only three asymptotic regimes: (1)minimality (all trajectories are dense); (2) there exist a global attractor anda global repellor, which are periodic and might coincide; (3) there exists abeam of periodic trajectories, whose boundary (if any) comprises an attractorand a repellor for all the other trajectories. Furthermore, in the prominentcase where the table is an actual trapezoid, we study the sets in parameterspace relative to the three regimes. We prove in particular that the set for(1) is a positive-measure fractal; the set for (2) has positive measure (giving arigorous proof of the existence of Arnol’d tongues for internal-wave billiards);the set for (3) has measure zero.
Mathematics Subject Classification (2020):
Primary: 37N10, 37C83,37E05. Secondary: 37E45, 37C70, 37C25, 76B55.
Keywords:
Internal-wave billiards, periodic attractors, circle homeomor-phisms, Arnol’d tongues, devil’s staircase. ∗ Dipartimento di Matematica, Universit`a di Bologna, Piazza di Porta San Donato 5, 40126Bologna, Italy. E-mail: [email protected] . † Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy. ‡ Dipartimento di Matematica, Universit`a di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy.E-mail: [email protected] . § Dipartimento di Matematica e Applicazioni, Universit`a di Milano - Bicocca, Via Roberto Cozzi55, 20125 Milano, Italy. E-mail: [email protected] . a r X i v : . [ m a t h . D S ] F e b Lenci, Bonanno, Cristadoro
When an incompressible fluid is stably stratified by a linear increase of density inthe direction of gravity, a periodic perturbation can generate gravity waves whosedirection of propagation is dictated only by the frequency of the forcing [19]. These internal waves appear ubiquitously in oceans and in the atmosphere, where theyplay a key role in mixing processes and energy dissipation [18]. When internalwaves encounter a rigid body, they are reflected according to this non standard rule[13, 17]:(a) the angles that the incident and reflected velocities make with the verticaldirection are equal or supplementary (in other words, the two vectors belongto the boundary of the same vertical double cone);(b) the orthogonal projections of the incident and reflected velocities on the normalto the body at the reflection point are opposite to each other;(c) the orthogonal projections of the incident and reflected velocities on the direc-tion orthogonal to both said normal and the vertical direction are the same.Observe that the standard (Fresnel) reflection rule is the same as the above exceptthat (a) is replaced by the condition that the angles of incidence and of reflection relative to the normal are the same. It can be checked that with the above rule,which we refer to as the internal-wave reflection rule , the moduli of the incident andreflected velocities are in general not the same [17, Sect. 2.2] — with the Fresnelrule they are.It is also easy to see that the internal-wave reflection rule causes a beam ofparallel rays to expand and/or contract its section, in general, at every reflection.In their seminal paper [15], Mass and Lam predicted that, in a closed container,the contraction effect may prevail in certain parts of phase space so as to generatea periodic attractor for the waves. This internal wave attractor , as it was dubbed,is determined solely by the geometry of the container and the frequency of theperiodic perturbation. This prediction was soon confirmed experimentally [14] andlater other laboratories [9, 8, 7, 1] and numerical simulations [6, 2] verified andextended the findings of [14]. (Here and in the rest of this paper we do not pretendto cite the entire literature on the subject of internal waves, which is massive: weonly reference the publications that are closest to our work.)From a theoretical point of view, the first step towards understanding the struc-ture of internal waves in closed containers consists in studying its ray dynamics [15, 14, 16, 13, 12, 9, 8, 17, 3], which is defined by the free motion of a point particleinside the container, subject to the internal-wave reflection law at its boundary. Thisdynamics has been studied mostly in two-dimensional domains, which one thinksof as vertical, relative to gravity. In this case the velocity of the particle can onlyassume four directions. If we indicate a direction by means of the angle ϑ it formswith the direction of gravity, the four possibilities are: θ, π − θ, π + θ, − θ , where θ nternal-wave billiards θ will play the equivalent role of a certain initial direction ofthe motion, see below.)We denote the domain by Ω and assume that its boundary ∂ Ω is piecewisesmooth. When the particle hits a point P ∈ ∂ Ω, its velocity changes instantaneouslyand takes the unique direction, among the 4 possibilities, that points towards theinterior of Ω and is not the opposite of the incoming direction. (This rule is am-biguous when the slope of ∂ Ω at P is either undefined or in one of the four specialdirections. We discard these trajectories, which amount to a Lebesgue zero-measureset in four-dimensional phase space.) As for the modulus of the outgoing velocity,this is a function of the incoming velocity and the slope of ∂ Ω at P [17, Sect. 2.2]:we do not recall it here because it will soon become irrelevant. Thus, there are twotypes of reflections: vertical reflections, where the direction of the velocity turnsfrom a given ϑ to − ϑ , and horizontal reflections, where the direction turns from ϑ to π − ϑ .We call internal-wave billiard any system like the above, with the difference thatthe speed of the particle, i.e, the modulus of the velocity, is constantly equal to 1 .This reparametrization of time naturally changes the dynamical system, so the raydynamics in Ω is not the internal-wave billiard in Ω, but if one is only interested incombinatorial/topological properties (say, periodicity or density of the trajectories,attractors, etc.), the two systems are equivalent. In the rest of the paper we onlydeal with internal-wave billiards, referring to Ω as the (billiard) table .Different tables have been investigated [15, 12, 16], showing the ubiquity ofinternal wave attractors, but the rectangular trapezoid, which was the shape chosenin the first actual experiment with internal waves [14], emerged as a prototypicalexample. When Ω is a rectangular trapezoid with horizontal bases and a verticalleg, reflections are always of horizontal type at the bases and of vertical type at thevertical leg. As for the slanted leg, let us look at Fig. 1 (b) to define the angles α and θ , the latter denoting the initial direction of the particle, conventionally takenat the vertical leg (we may also assume θ ∈ [0 , π/ < θ < α and vertical for α < θ < π/
2. We do not consider the cases θ ∈ { , α, π/ } becausethey are formally ill-defined and/or trivial. The case 0 < θ < α is also simple: alltrajectories converge to the rightmost corner of the trapezoid. It is for α < θ < π/ Lenci, Bonanno, Cristadoro
Figure 1:
Examples of billiard tables Ω. tables which includes the rectangular trapezoid. Specifically, we allow the slantedleg of the trapezoid to be replaced by a piecewise smooth curve with downwardconcavity; see Fig. 1 for examples of tables in this class and Section 2 for a precisedefinition of it. Our main result gives a complete description of the asymptotic be-havior of the system. In particular, it is proven that there are only three possibilities,with subcases:1. All trajectories are dense.2. There exist a unique global attractor and a unique global repellor, which areperiodic with the same number of bounces off the boundary of the table, andmight possibly coincide.3. There exists a beam of periodic trajectories.(a) If the beam is non-degenerate, its boundary consists of two periodic tra-jectories, which are, respectively, the attractor and the repellor of alltrajectories outside of the beam; otherwise(b) the beam may reduce to a single periodic trajectory, which acts as botha global attractor and a global repellor; or(c) the beam may comprise all trajectories, resulting in the case where alltrajectories are periodic with the same number of bounces off the bound-ary.Furthermore, if the slanted side of Ω does not contain any segment, case 3 coin-cides with subcase (b); if the slanted side of Ω is a segment (i.e., Ω is a rectangulartrapezoid), case 3 coincides with subcase (c). All these statements are direct conse-quences of Theorem 2.2, which describes the dynamics of a suitable Poincar´e mapfor the billiard. Section 2 is devoted to introducing, stating and proving Theorem2.2. nternal-wave billiards
Acknowledgments.
We thank Thierry Dauxois for useful discussions in the earlystage of this work. The present research was partially supported by the PRIN Grant2017S35EHN, MUR, Italy. It is also part of the authors’ activity within the UMIGroup
DinAmicI and the Gruppo Nazionale di Fisica Matematica, INdAM.
We study the internal-wave billiard in a table Ω that is a generalization of a rectan-gular trapezoid, like the examples shown in Fig. 1. In suitable units, Ω has height1/2 and is completely specified by the position and shape of its rightmost boundary ∂ R Ω, which we assume to be the graph of a piecewise C , strictly decreasing, con-cave function, possibly with the addition of a vertical segment attached to its lowerend. Fig. 1 (a) defines three important parameters for the dynamics, the angles α m , α M and θ . In particular, θ ∈ (0 , π/ ∪ ( π/ , π ) is the particle’s initial direction,having assumed (without loss of generality) an initial position on ∂ L Ω, the verticalside of Ω. We do not consider horizontal or vertical initial directions for the reasonsexplained earlier.If θ ∈ (0 , α m ) ∪ ( π − α m , π ), it is easily verified that all reflections at ∂ R Ωare horizontal (as per the definition given in the introduction), implying that everytrajectory converges to the lower endpoint of ∂ R Ω. If θ ∈ [ α m , α M ] ∪ [ π − α M , π − α m ], ∂ R Ω is generally split into two parts, which give rise to horizontal and verticalreflections, respectively. This is the more complicated case, which we do not studyat the present time, as we intend to keep our mathematical machinery to a minimum.Moreover, when Ω is a trapezoid, which a case of interest here, the dynamics for θ = α is ill-defined.In this paper we restrict to the case θ ∈ ( α M , π/ ∪ ( π/ , π − α M ) to ensurethat all reflections at ∂ R Ω are vertical. Since the particle’s velocity can only assume4 values, it is convenient to represent the dynamics on Ω as a linear flow on Ω ,the 4-fold copy of Ω represented in Fig. 2 as a subset of R . The linear flow onΩ is defined as a fixed-velocity motion on it, with the provision that a trajectoryhitting a boundary point continues on the opposite boundary point with the samevelocity. Two boundary points of Ω are said to be opposite to each other if theybelong to the upper/lower boundaries of Ω and have the same abscissa, or to theleft/right boundaries and have the same ordinate. Fig. 2 also displays a trajectory Lenci, Bonanno, Cristadoro of the linear flow, together with its projection on Ω. The defining parameter of thelinear flow is the angle θ introduced earlier, which we also call the direction of theflow. We convene that the speed of the flow is 1. It is apparent that the naturalprojection Ω −→ Ω maps trajectories of the flow into trajectories of the billiard.The procedure whereby one passes from the internal-wave billiard on Ω to the linearflow on Ω is also called billiard unfolding . (This technique has been applied veryfruitfully to polygonal (ordinary) billiards; see, e.g., the reference list of [4].) Figure 2:
A trajectory of the linear flow in Ω , together with thecorresponding orbit for the Poincar´e map f : Σ −→ Σ. The definitionof x c ( x ), cf. Proposition 2.1, is also illustrated. Our assumptions on the billiard table Ω, given at the beginning of this section,can be restated as assumptions on Ω as follows: the upper boundary of Ω is thegraph of a function b : [ − / , / −→ R + , which is even, piecewise C , concave,and (not necessarily strictly) decreasing on [0 , / α ( x ) := arctan( b (cid:48) ( x )). Thisexpression fails to be defined in at most countably many points of ( − / , / α is a (not necessarily strictly) decreasing odd function with α ( − /
2) = α M and α (1 /
2) = − α M . In particular, when b (cid:48) ( x ) is defined, x ≤ α ( x ) ≥ x ≥ α ( x ) ≤ , which we henceforth do, we regardΩ as a surface homeomorphic to a torus. The metric of R induces on this surface nternal-wave billiards
7a metric which is defined and flat everywhere except for a closed curve (which, intopological terms, is a simple, non-contractible loop). The linear flow is the geodesicflow for this metric, subject to a non-isometric identification rule between the twosides of the closed curve.A convenient way to study the elementary properties of a flow is by means of asuitable Poincar´e map. In our case, a good choice is to take the first-return mapto the horizontal segment Σ shown in Fig. 2 (a) . We denote it f : Σ −→ Σ. Asper our boundary identifications, Σ has the topology of a circle. In any case, withthe units that we have chosen, its length is 1. Observe that points on this Poincar´esection correspond to the particle being on ∂ L Ω with velocity directed as θ or π − θ .In other words, Σ is in 2-to-1 correspondence with the set of initial positions wechose for our dynamics. This fact and the symmetry of Ω show that it is no loss ofgenerality to restrict the directions of the linear flow to θ ∈ ( α M , π/ S := R / Z in the natural way, we also regard f as a homeomorphism of S . Lemma 2.1
Let us denote by x c ( x ) the abscissa of the first collision point of thetrajectory starting form x ∈ Σ with the upper boundary of Ω (see Fig. 2(a)). Forall x ∈ Σ such that α ( x c ( x )) is defined, we have f (cid:48) ( x ) = sin( θ + α ( x c ( x )))sin( θ − α ( x c ( x ))) . Proof.
This proof will be illustrated by Fig. 2 (b) . Given x ∈ Σ such that α ( x c ( x ))is well defined in a neighborhood of x , let us take δx so small that such neighborhoodcontains [ x, x + δx ] (having chosen δx > x, x + δx ] has width δw = δx sin θ and thusprojects on the upper boundary of Ω an arc of length δr = δw sin( θ − α ) + o ( δx ) = δx sin θ sin( θ − α ) + o ( δx ) , as δx → . (2.1)Here, with a slight abuse of notation, we have denoted α := α ( x c ( x )). The beamthen continues from the opposite arc, on the lower boundary of Ω . The lengthof this arc is clearly δr (cid:48) = δr but, reversing the previous reasoning, its width is δw (cid:48) = δr (cid:48) sin( θ + α ) + o ( δx ). The first intersection of this beam with Σ is thesegment [ f ( x ) , f ( x + δx )], whose length is f ( x + δx ) − f ( x ) = δw (cid:48) sin θ = δx sin( θ + α )sin( θ − α ) + o ( δx ) , as δx → . (2.2)Dividing the above by δx and taking the limit proves the proposition. Q.E.D.
Another property of f is that, for x, x ∈ Σ, x = f ( x ) ⇐⇒ − x = f ( − x ) . (2.3) Lenci, Bonanno, Cristadoro
In other words, the graph of f , as represented in R , is symmetric around thebisectrix of the second and fourth quadrants. This is easily seen by drawing a first-return segment of a trajectory of the linear flow on Ω and rotating Ω by 180degrees, equivalently, by exploiting the fact that the billiard dynamics commuteswith time-inversion.In order to study the topological/combinatorial properties of the internal-wavebilliard, it suffices to study the corresponding properties of f on S . There existsa well-developed theory of circle homeomorphisms, dating back from Poincar´e, whointroduced the notion of rotation number, which we briefly recall. Let F : R −→ R be a lift of f , i.e., a homeomorphism of R which is well defined on the equivalenceclasses of R / Z (this turns out to be the same as the property F ( x + k ) = F ( x ) + k ,for all x ∈ R and k ∈ Z ) and acts like f there. Then the quantity ρ ( f ) := lim n →∞ F n ( x ) − xn (2.4)does not depend on x . If taken mod 1, as is customary, it is also independent of thechoice of F . One calls ρ ( f ) the rotation number of f . In this paper, unless otherwisestated, we will always represent it as a number in [0 , all the orbits of f . Notation.
In what follows we will need to use cyclic indices , namely, elements of Z q := Z /q Z , for some q ∈ Z + . We can think of them as elements of { , , . . . , q − } with the understanding that the sum of any such element with an integer is intendedmod q . Theorem 2.2
Let ρ ( f ) ∈ [0 , denote the rotation number of f : S −→ S .(i) If ρ ( f ) (cid:54)∈ Q , f is topologically equivalent (i.e., conjugated via homeomorphism)to the rotation x (cid:55)→ x + ρ ( f ) (mod 1) on S . In particular, f is minimal, i.e.,all orbits are dense.(ii) If ρ ( f ) ∈ Q , we write ρ ( f ) = p/q , where, if ρ ( f ) > , p and q are coprimepositive integers, or else p = 0 and q = 1 . There exist two sets P ± := q − (cid:91) i =0 P ± i , where the P ± i are closed intervals. All points in P ± are q -periodic and suchthat, for all i ∈ Z q and n ∈ N , f n (cid:0) P ± i (cid:1) = P ± i + pn . All points in S \ ( P + ∪ P − ) are non-periodic.Only two possibilities are given: nternal-wave billiards (a) Each P ± i reduces to a point, which we denote x ± i . These points are orderedas follows, according to the orientation of S : x − ≤ x +0 ≤ x − ≤ x +1 ≤ · · · ≤ x − q − ≤ x + q − ≤ x − . So O ± := P ± = { x ± , x ± , . . . , x ± q − } are two (possibly coinciding) periodicorbits such that f n ( x ± i ) = x ± i + pn , for all i ∈ Z q and n ∈ N . As for theother points: ∀ x ∈ (cid:0) x − i , x + i (cid:1) , f n ( x ) ∈ (cid:0) x − i + pn , x + i + pn (cid:1) and lim n → + ∞ f nq ( x ) = x + i , lim n →−∞ f nq ( x ) = x − i ; ∀ x ∈ (cid:0) x + i , x − i +1 (cid:1) , f n ( x ) ∈ (cid:0) x + i + pn , x − i +1+ pn (cid:1) and lim n → + ∞ f nq ( x ) = x + i , lim n →−∞ f nq ( x ) = x − i +1 . Hence O + , respectively O − , is the global attractor, respectively repellor,of the system.(b) All P ± i are non-degenerate intervals, with P + i = P − i , for all i ∈ Z q .Denoting any such interval [ x Li , x Ri ] := P + i = P − i , we have x L < x R ≤ x L < x R ≤ · · · ≤ x Lq − < x Rq − ≤ x L , with the property that x Ri < x Li +1 holds for some i ∈ Z q if, and only if, itholds for all i ∈ Z q .(1) If x Ri < x Li +1 for all i then, for all i ∈ Z q and n ∈ N , f n ( P ± i ) = P ± i + pn and ∀ x ∈ (cid:0) x Ri , x Li +1 (cid:1) , f n ( x ) ∈ (cid:0) x Ri + pn , x Li +1+ pn (cid:1) and lim n → + ∞ f nq ( x ) = x Ri , lim n →−∞ f nq ( x ) = x Li +1 . Therefore O + := { x R , x R , . . . , x Rq − } and O − := { x L , x L , . . . , x Lq − } are, respectively, the unique (but not global) attractor and repellor ofthe system.(2) If x Ri = x Li +1 for all i , all orbits are periodic.Under the additional assumption that ∂ R Ω does not contain any segment (equiv-alently, the function b whose graph is the upper boundary of Ω is strictlyconcave), case (b) cannot occur. Remark 2.3
Since an orbit of f is dense/periodic/attracting/repelling if, and onlyif, the corresponding flow trajectory in Ω is, all the statements of Theorem 2.2are immediately translated to statements about the internal-wave billiard in Ω. Inparticular, this proves the claim made in the introduction about the three solepossibilities for the asymptotics of the trajectories.0 Lenci, Bonanno, Cristadoro
Proof of Theorem 2.2.
Poincar´e’s classical theory of circle homeomorphisms[5, Sect. I.1] states that, for any orientation-preserving f : S −→ S , the followingdichotomy holds:1. If no periodic orbit exists, then f is topologically semiconjugate to an irrationalrotation [5, Thm. 1.1], which must necessarily be the rotation by ρ ( f ) (cid:54)∈ Q .2. If a periodic orbit exists and q is its (primitive) period, then necessarily ρ ( f ) = p/q , where p is either 0 or coprime to q . Let x , x , . . . , x q − be an orientation-preserving labeling of the points of the periodic orbit. By definition of rotationnumber, for all i ∈ Z q and n ∈ N , one has f n ( x i ) = x i + pn (2.5)(recall the convention on cyclic indices). Therefore f n ([ x i , x i +1 )) = [ x i + pn , x i +1+ pn ) . (2.6)Necessarily, then, if x (cid:48) is another periodic point, its combinatorics is the sameas that of x , that is, its period is q and, for any orientation-preserving labeling x (cid:48) , x (cid:48) , . . . , x (cid:48) q − of its orbit, the analogue of (2.5) holds.To establish (i) it suffices to verify that, for our particular f , the topologicalsemiconjugacy of case 1 is in fact a topological conjugacy. This is exactly theassertion of Denjoy’s Theorem for certain circle diffeomorphisms [5, Sect. I.2]. Asexplained in [5, Rmk on p. 38], Denjoy’s Theorem also holds for homeomorphisms f that are piecewise differentiable and such that log | f (cid:48) | can be extended to a mapwith bounded variation. Our f falls in this category, as shown momentarily.Let us call break point of f (cid:48) any x ∈ S such that∆ f (cid:48) ( x ) := lim s → x + f (cid:48) ( s ) − lim s → x − f (cid:48) ( s ) (cid:54) = 0 , (2.7)provided the limits exist. A break point x of f (cid:48) is said to be of increase or decrease if ∆ f (cid:48) ( x ) > f (cid:48) ( x ) <
0, respectively.
Lemma 2.4
The homeomorphism f defined earlier is such that lim s → x ± f (cid:48) ( s ) existsat all x ∈ S ; f (cid:48) has exactly one break point of increase, denoted a , and at mostcountably many break points of decrease, denoted { a i } i ≥ . This implies that f (cid:48) iscontinuous on S \ { a i } i ≥ with positive one-sided limits everywhere. Furthermore, f is concave on the arc S \ { a } . Finally, log f (cid:48) can be extended to a map withbounded variation. Proof of Lemma 2.4.
By construction of the function b : S −→ R + intro-duced earlier, b (cid:48) has exactly one break point of increase ¯ a (with ¯ a = − /
2, inthe identification S ∼ = [ − / , / nternal-wave billiards { ¯ a i } i ≥ . In all other points of S , b (cid:48) is continuous. Also, b (cid:48) is decreasing(not necessarily strictly) on S \ { ¯ a } .On the other hand, Lemma 2.1 states that f (cid:48) = g θ ◦ arctan ◦ b (cid:48) ◦ x c , where g θ ( α ) := sin( θ + α )sin( θ − α ) (2.8)defines a strictly increasing continuous function [ − α M , α M ] −→ R + and x c : S −→ S is the map defined in the statement of Lemma 2.1, which is easily seen to be anorientation-preserving homeomorphism. Therefore, there is a bijective correspon-dence between the break points of b (cid:48) and those of f (cid:48) , such that the two functionshave the same monotonicity properties between corresponding pairs of break points.This proves all the assertions of Lemma 2.4, except for the last one.As for the last assertion, let us observe that, by definition, the function f (cid:48) isnot defined at its break points, so let us extend it to the whole of S by setting f (cid:48) ( a i ) := lim s → a + i f (cid:48) ( s ) (having employed the common abuse of notation wherebythe extension has the same name as the extended function). It is evident thatVar S \{ a } (log f (cid:48) ) = ∆ log f (cid:48) ( a ), whence Var S (log f (cid:48) ) = 2 ∆ log f (cid:48) ( a ) < ∞ . (HereVar S \{ a } ( · ) is the variation of a real-valued function on the arc S \{ a } and Var S ( · )is the variation on the whole torus, amounting to the former variation plus thevariation at a .) Q.E.D.
Now for the statements (ii) of Theorem 2.2. By Poincar´e’s dichotomy, if ρ ( f ) = p/q as in (ii) , f has at least a periodic orbit of cardinality q , that is, f q has at least q fixed points. The previous proposition and the identity( f q ) (cid:48) ( x ) = q − (cid:89) k =0 f (cid:48) ( f k ( x )) (2.9)show that ( f q ) (cid:48) has at most q break points of increase (namely { f − k ( a ) } q − k =0 , keepingin mind that some of these points may coincide), outside of which f q is piecewise C and concave. For the rest of this proof we refer to this property as the ‘concavityproperty of f q ’. Identifying S with [ − / , / f q can only look likeone of the cases depicted in Fig. 3.In particular, the set of fixed points of f q comprises at most 2 q , possibly de-generate, closed intervals P ± i . By the concavity property of f q , the number of suchintervals is 2 q if and only if each P ± i is a singleton { x ± i } . In this case we label x − i ,respectively, x + i , the repelling, respectively, attracting fixed points. Since, relativeto the orientation of S , they alternate, the labelling can be done so that x − < x +0 < x − < x +1 < · · · < x − q − < x + q − < x − . (2.10)Thus, by the second part of Poincar´e’s dichotomy, f n ( x ± i ) = x ± i + pm , for all i ∈ Z q and n ∈ N . This the case of Fig. 3 (a) .2 Lenci, Bonanno, Cristadoro (a) (b) (c)(d) (e)
Figure 3:
Possible graphs of the map f q , when a q -periodic orbit of f exists. If the graph of f q touches the bisectrix of the first and third quadrants in a singlepoint of abscissa x then, if the graph touches “from below” (Fig. 3 (b) ), x is a fixedpoint of f q which is repelling on the left and attracting on the right; if the graphtouches “from above” (Fig. 3 (c) ), x is attracting on the left and repelling on the right.The same must therefore happen for all points of the f -periodic orbit { f i ( x ) } i ∈ Z q ,which must be distinct, by part 2 of Poincar´e’s dichotomy. The concavity propertyof f q shows that each of these points must belong to one and only one concavepart of f q , proving that there can be no other periodic orbits of f . Following theirorientation on S , we label x − , x − , . . . , x − q − the points of the aforementioned periodicorbit. In the case of left-repelling and right-attracting points (Fig. 3 (b) ), we also set x + i := x − i ; in the case of left-attracting and right-repelling points (Fig. 3 (c) ), we set x + i := x − i +1 . In either case, we finally denote P ± i := { x ± i } .The above considerations prove all the assertions of Theorem 2.2 (ii)(a) , whenthe sets P ± i are singletons. If P + ∪ P − is not made up of isolated points, then itmust include a closed interval, as we have seen. Take the largest interval (or, incase of a tie, one of the largest intervals) within P + ∪ P − . There can only be twocases: either this closed interval is a proper subset of S , in which case, withoutloss of generality, we denote it P +0 := [ x L , x R ]; or it is the whole of S , in whichcase we choose any point x L ∈ S and set x R := f ( x L ), P +0 := [ x L , x R ]. In theformer case, since f is a homeomorphism and preserves the property of being or notan f q -periodic point, we see that the sets { f i ( P +0 ) } i ∈ Z q are pairwise disjoint, closedintervals; they are also distinct, since all f -periodic orbits must have period q , cf.Fig. 3 (d) . The concavity property of f q shows that there can be no more periodic nternal-wave billiards { f i ( x L ) } i ∈ Z q are distinct, we have thatthe sets [ f i ( x L ) , f i ( x R )], i ∈ Z q , cover S and intersect only at their endpoints. SeeFig. 3 (e) .In either case, we denote P +0 , P +1 , . . . , P + q − the intervals { f i ( P +0 ) } i ∈ Z q , orderedaccording to the orientation of S , and P − i := P + i for all i . The above facts proveall the claims of part (ii)(b) of the theorem. In particular, Fig. 3 (d) corresponds tocase (ii)(b)(1) and Fig. 3 (e) corresponds to case (ii)(b)(2) .Lastly, observe that if the function b is strictly concave, (2.9) shows that f q isstrictly concave in each of its concavity interval, whence P + ∪ P − can only containisolated points, forcing the case (ii)(a) . Q.E.D.
Remark 2.5
Theorem 2.2 has implications on the ergodic properties of f and thusof the billiard flow. Here we only mention a few facts about f , whence it is easy toderive the corresponding statements for the flow. If ρ ( f ) (cid:54)∈ Q , there exists a uniqueinvariant probability measure µ on S such that, for all continuous ϕ : S −→ R andall x ∈ S , lim n →±∞ n n − (cid:88) k =0 ϕ ( f k ( x )) = (cid:90) S ϕ dµ. (2.11)(The fact that, for a continuous observable, the time averages over all — not justalmost all — orbits converge to the same limit is peculiar to this case, where uniqueergodicity holds.) It is worth mentioning that µ need not be mutually absolutelycontinuous with the Lebesgue measure on S . In fact, in the case of the trapezoidalbilliard, where the map f is piecewise linear with two break points of f (cid:48) (see Section3), we know by [10, Thm. 7.4] that µ is singular. If instead ρ ( f ) ∈ Q then, for allcontinuous ϕ : S −→ R and all x (cid:54)∈ P + ∪ P − ,lim n →±∞ n n − (cid:88) k =0 ϕ ( f k ( x )) = (cid:90) S ϕ du O ± , (2.12)where u O ± are the uniform probability measures, respectively, on the periodic attrac-tor O + and repellor O − . Recall that, in case (ii)(a) of Theorem 2.2, S \ ( P + ∪P − ) = S \ ( O + ∪ O − ). In this section we consider the case where Ω is the rectangular trapezoid of Fig. 1 (b) .We improve the general results of Theorem 2.2 and study the sets of parameters forwhich each of the only possible three cases happens: (1) all orbits are dense; (2) thereexist a global attractor and a global repellor; (3) all trajectories are periodic withthe same period. We shall see in particular that cases (2) and (3) occur when therotation number, in lowest terms, has an odd or an even denominator, respectively.4
Lenci, Bonanno, Cristadoro
We recall that the height of the trapezoid was fixed to 1/2 and that in this casethe two equal angles α M = α m are denoted α . Let us fix the length of the shorterbase to a certain value (cid:96) > θ ∈ ( α, π/ f : S −→ S introduced in Section2. Lemma 2.1 shows that the derivative f (cid:48) only takes the values Λ and Λ − , whereΛ := sin( θ + α )sin( θ − α ) > . (3.1)In order to simplify the ensuing computations, we impose the extra condition θ ≥ arctan(2 (cid:96) ). The left out values of θ , if any, are qualitatively the same as the onesdescribed below. Recalling that the graph of f is symmetric around the bisectrixof the second and fourth quadrants of the square [ − / , / , we conclude that itmust look like the one shown in Fig. 4. More in detail, using the notation of Lemma2.4, let us call a and a , respectively, the break points of increase and decrease of f (cid:48) . Simple calculations based on Fig. 2 show that a = tan θ − (cid:96) θ ∈ (cid:18) , (cid:19) , a = − (cid:96) + tan α θ ∈ (cid:18) − , (cid:19) , (3.2)and f ( a j ) = − a j , for j = 0 , Figure 4:
The graph of f in the case where Ω is a rectangular trapezoid. Instead of studying f directly, it will be convenient to consider its lift (cf. Section2) F : R −→ R uniquely defined by the conditions f ( x ) = F ( x ) − (cid:22) F ( x ) + 12 (cid:23) , ∀ x ∈ (cid:20) − , (cid:19) ; (3.3) F ( a ) = − a . (3.4) nternal-wave billiards F ( a ) = 1 − a . The last two equalities, togetherwith the knowledge of the slope of F before and after a and a , allow us to derivethe following expression for F : F ( x ) = − a + Λ − ( x − a ) , if x ∈ [ a , a ];1 − a + Λ( x − a ) , if x ∈ [ a , a + 1]; F ( x + k ) − k, if x + k ∈ [ a , a + 1] with k ∈ Z . (3.5)We now improve on Theorem 2.2 giving more precise statements in the case ofrational rotation numbers. Proposition 3.1
Let f : S −→ S be as defined above. Assume ρ ( f ) = p/q , with p, q coprime positive integers, or p = 0 , q = 1 . If q is even then f q = Id , that is,all orbits are periodic with primitive period q ; moreover, a and a are in the sameperiodic orbit. If q is odd then f q (cid:54) = Id ; also a and a are not in the same periodicorbit. Proof.
We first consider the case where q is even. Let x ∈ S be a periodic pointwith (primitive) period q and denote by x , x , . . . , x q − be the points of its orbit,labeled according to the orientation of S .By absurd, we assume that a and a are not periodic points. As explained inthe proof of Theorem 2.2, in this case f q is piecewise linear with 2 q break points of( f q ) (cid:48) , which correspond to the points in the backward orbits of a and a up to time − q + 1. In other words, since a and a are not periodic points, the graph of f q fallsin the case of Fig. 3 (a) ; more precisely, it is a polyline made up of 2 q segments eachof which crosses the bisectrix of the first and third quadrants. So, for all i ∈ Z q ,there are exactly two break points between x i and x i +1 : one in the orbit of a andone in the orbit of a (recall that i is a cyclic index, whence x q ≡ x ).For all points x ∈ S such that ( f q ) (cid:48) ( x ) exists, we can write ( f q ) (cid:48) ( x ) = Λ n + − n − ,where n + = n + ( x ) and n − = n − ( x ) are the number of times the orbit of x up totime q − f (cid:48) is Λ and Λ − , respectively. Clearly, n + + n − = q ,which is even, so n + − n − is also even. As x varies from x to x , ( f q ) (cid:48) ( x ) varies atthe two break points in the interval ( x , x ): it is immediate to verify that passingthrough the point in the orbit of a the effect is that n + increases by one and n − decreases by one, and the opposite happens passing through the point in the orbit of a . Therefore, if ( f q ) (cid:48) ( x ) ≥ Λ , then ( f q ) (cid:48) ( x ) ≥ x ∈ ( x , x ] except for twobreak points. The last two inequalities are in contradiction with f q ( x ) = x and f q ( x ) = x . Analogously, if ( f q ) (cid:48) ( x ) ≤ Λ − , then ( f q ) (cid:48) ( x ) ≤ x ∈ [ x , x ]except for two break points, again a contradiction. Thus, ( f q ) (cid:48) ( x ) = 1. But thisimplies that ( f q ) (cid:48) ( x ) = 1 for all x ∈ [ x , ¯ a ), where ¯ a denotes the first break pointto the right of x . All such points, then, including ¯ a , are periodic, which is acontradiction, because ¯ a is in the orbit of a or a .We conclude that a or a must be periodic points, that is, we are in the cases (b), (c) or (e) of Fig. 3 (case (d) is not achievable with a polyline of 2 q segments).6 Lenci, Bonanno, Cristadoro
Let us assume for now that there exist periodic break points of ( f q ) (cid:48) , that is, we arenot in case (e) . Let x be a periodic break point. One can use essentially the samearguments as in the previous paragraph to prove that ( f q ) (cid:48) ( x + ε ) = 1 for all small ε >
0. (In fact, it is not possible that x and x are periodic points, ( f q ) (cid:48) ( x ) ≥ x in ( x , x ), and ( f q ) (cid:48) ( x + ε ) > ε >
0; the same goes forthe analogous statement with reversed inequalities.) But this is incompatible withcases (b) and (c) of Fig. 3.So only case (e) is possible, i.e., f q = Id. This implies in particular that a and a belong to the same periodic orbit. In fact, if not, ( f q ) (cid:48) would have a positivejump at a , contradicting ( f q ) (cid:48) ≡ q is odd. In this case ( f q ) (cid:48) ( x ) (cid:54) = 1 forall x , since there is no choice for n + and n − defined above to satisfy n + = n − and n + + n − = q . One also sees that a and a are not in the same periodic orbit,otherwise there would be no break points of ( f q ) (cid:48) . Q.E.D.
We now use the theory of rotation numbers for circle homeomorphisms togetherwith Proposition 3.1 to show that there is an abundance of parameters for whichthe rotation number is irrational. In the following, to avoid more involved butno more instructive computations, we further restrict θ to belong to the interval[arctan(2 (cid:96) + tan α ) , π/ θ are enough to explore the whole rangeof rotation numbers in the interval (0 , t := 1 / tan θ in place of θ . Theorem 3.2
For t ∈ (0 , t M ) , with t M := (2 (cid:96) + tan α ) − , denote by f t : S −→ S the piecewise linear circle homeomorphism defined by (cid:96) , α and θ = arctan(1 /t ) , asshown above. Also, denote by F t : R −→ R its lift as in (3.5). Then the function t (cid:55)→ ρ ( f t ) is a Lipschitz increasing devil’s staircase (i.e., a Lipschitz, increasing, non-constant function that is constant on each interval of a family with dense union) andits range is (0 , . Furthermore, m ( { t ∈ (0 , t M ) : ρ ( f t ) (cid:54)∈ Q } ) > , where m is the Lebesgue measure on R . Proof.
The parameters used in (3.5) can be written as functions of t as follows:Λ( t ) = 1 + t tan α − t tan α , a ( t ) = 12 − t(cid:96) , a ( t ) = − t (cid:18) (cid:96) + 12 tan α (cid:19) . (3.6)The properties of the above three functions of t enable us to prove an importantlemma. Lemma 3.3
The collection ( f t ) t ∈ (0 ,t M ] is a continuous strictly increasing family ofcircle homeomorphisms, i.e., for every t ∈ (0 , t M ] there exists a lift F t : R −→ R of f t , such that t (cid:55)→ F t is continuous in the sup norm of R and, for all x ∈ R , t (cid:55)→ F t ( x ) is strictly increasing. nternal-wave billiards Proof of Lemma 3.3.
The collection of lifts F t defined by (3.5) and (3.6) doesthe job. The continuity of t (cid:55)→ F t in the sup norm is a direct consequence of thecontinuity of the parameters in (3.6). As for the monotonicity, observe that a ( t )and a ( t ) are strictly decreasing functions of t . On the other hand, by construction, F t ( a ( t )) = − a ( t ) and F t ( a ( t )) = 1 − a ( t ). Since F t is a lift, F t ( a ( t ) + k ) = k − a ( t ) , F t ( a ( t ) + k ) = k + 1 − a ( t ) , (3.7)for all k ∈ Z . This means that, in the plane with Euclidean coordinates ( x, y ), thepoints ( a ( t ) + k, k − a ( t )) and ( a ( t ) + k, k + 1 − a ( t )) belong to the graph of F t .But they also belong, respectively, to the lines y = 2 k − x and y = 2 k + 1 − x , seeFig. 5. As t increases, these points slide to the left and up along those lines. But thegraph of F t is the infinite polyline interpolating all such points, in the right order.This shows that, for all t < t , the graph of F t lies strictly below the graph of F t ,which is equivalent to the last assertion of Lemma 3.3. Q.E.D.
Figure 5:
Graphs of F t for three values of t . The black graph corre-sponds to f as in Fig. 4. The short dashed lines are the lines y = − x and y = − x + 1 in coordinates ( x, y ). The above lemma is key here because it is a known fact that the function f (cid:55)→ ρ ( f ) is continuous and increasing in the space of orientation-preserving cir-cle homeomorphisms (meaning that the r.h.s. of (2.4), as an element of R , is a8 Lenci, Bonanno, Cristadoro continuous function of F , w.r.t the sup norm, which does not decrease if F is in-creased pointwise; more details in [11, § t (cid:55)→ ρ ( f t ) is continuous andincreasing. Moreover, by general results (see, e.g., [11, Prop. 11.1.9]), it is strictlyincreasing at irrational values, meaning that if ρ ( f t ) (cid:54)∈ Q then, for all sufficientlyclose t < t < t , ρ ( f t ) < ρ ( f t ) < ρ ( f t ).The next proposition is a corollary of Proposition 3.1. We will use it to proveTheorem 3.2, but the results are of independent interest as well. Proposition 3.4
Let ρ : (0 , t M ) −→ [0 , be defined by ρ ( t ) := ρ ( f t ) and denote by Q o the set of all rationals that can be written as p/q , with p and q coprime and q odd (by convention, / ∈ Q o ). Then:(i) for all r ∈ Q o , ρ − ( r ) is a non-degenerate closed interval and, for all t ∈ ρ − ( r ) , f t is not topologically equivalent to a rotation;(ii) for all r ∈ R \ Q o , ρ − ( r ) is a point t r and f t r is topologically equivalent to therotation by ρ ( f t r ) . Proof of Proposition 3.4.
Let us first observe that, since F t ( x ) is strictlyincreasing both in t and x , the graphs of F qt are strictly increasing in t , for all q ∈ Z + , i.e., ( f qt ) t ∈ (0 ,t M ) is also a strictly increasing continuous family of circlehomeomorphisms.If, for a given t r , ρ ( f t r ) = r ∈ Q o , Proposition 3.1 shows that f t r falls in the case (ii)(a) of Theorem 2.2 and so it cannot be conjugate to a rotation (there exist anattractor). Moreover, by the continuity of ( f qt ) t , a left and/or right perturbation in t preserves the q intersections between the graph of f qt and the bisectrix of the firstand third quadrants in [ − / , / . So ρ ( f t ) = r . This ends the proof of part (i) .As for part (ii) , we first consider r ∈ Q \ Q o and then r ∈ R \ Q . If ρ ( f t r ) = r ∈ Q \ Q o , Proposition 3.1 gives f qt = Id. By the strict monotonicity of f qt , anyarbitrarily small perturbation in t will produce a positive distance between the graphof f qt and the bisectrix of the first and third quadrants, making ρ ( f t ) arbitrarily closeto 0 or 1, but not equal to 0. Since it is a general fact that ρ ( f q ) = qρ ( f ) mod 1, itfollows that ρ ( f t ) must vary, whence ρ − ( r ) = { t r } . Moreover, it is a general fact thatif f q = Id then f is topologically equivalent to the rotation by ρ ( f ) = p/q , for some p . (Here is a sketch of its proof. Say that ρ ( f ) = p/q , with p coprime to q . Givena periodic orbit { x , x , · · · , x q − } , labeled according to the orientation of S , onedefines φ | [ x ,x ] to be any orientation-preserving homeomorphism [ x , x ] −→ [0 , /q ].Then, for x ∈ [ x i , x i +1 ], i ∈ Z q , one chooses n such that np = i mod q and defines φ ( x ) := φ ( f − np ( x )) + i/q . This gives a homeomorphism [ x i , x i +1 ] −→ [ i/q, ( i + 1) /q ].It is easy to verify that, for r ∈ [0 , φ ◦ f ◦ φ − ( r ) = r + p/q mod 1.)Finally, if ρ ( f t r ) = r ∈ R \ Q , the fact that ρ − ( r ) = { t r } follows from the strictmonotonicity of ρ at irrational values, as recalled earlier, and the fact that f t r istopologically equivalent to the corresponding rotation follows from Theorem 2.2 (i) . Q.E.D. nternal-wave billiards
19A further result in the theory of rotation numbers for circle homeomorphismsstates that, given a continuous monotonic family of orientation-preserving circlehomeomorphisms, such as our ( f t ) t ∈ (0 ,t M ) , if t (cid:55)→ ρ ( f t ) is non-constant and thereexists a dense set S ⊂ Q such that, whenever ρ ( f t ) ∈ S , f t is not topologicallyequivalent to a rotation, then t (cid:55)→ ρ ( f t ) is a devil’s staircase, cf. [11, Prop. 11.1.11].We can invoke this result by taking S := Q o as in the statement of Proposition3.4. Clearly, S is dense. It remains to show that the function ρ is non-constant.From the explicit expression of F t , see also Fig. 5, we know that there is a positivedistance between the graph of F t and the bisectrix of the first and third quadrantsin the plane. This implies that ρ ( t ) >
0, for all t >
0. On the other hand, since (cid:107) F t − Id (cid:107) ∞ vanishes, as t → + , thenlim t → + ρ ( t ) = 0 . (3.8)Thus, ρ is not constant and must be a devil’s staircase.To prove that Range( ρ ) = (0 ,
1) we employ the monotonicity of ρ together with(3.8) and lim t → t − M ρ ( t ) = 1 . (3.9)The above limit may be verified by means of (3.4) and the equality a ( t M ) = − / F t M ( − /
2) = 1 /
2, whence F nt M (cid:18) − (cid:19) = −
12 + n. (3.10)This shows that ρ ( f t M ) = 1 = 0 mod 1. But the other properties of ρ guaranteethat (3.9) is the only possibility.As for the Lipschitz property, we use the fact that for a map f with lift F = Id+Φone has ρ ( f ) = lim k →∞ k k − (cid:88) j =0 Φ( f k ( x )) , ∀ x ∈ R . (3.11)Therefore, for t , t ∈ (0 , t M ), | ρ ( f t ) − ρ ( f t ) | ≤ lim k →∞ k k − (cid:88) j =0 (cid:12)(cid:12) Φ t ( f k ( x )) − Φ t ( f k ( x )) (cid:12)(cid:12) ≤ (cid:107) Φ t − Φ t (cid:107) ∞ , (3.12)where Φ t := F t − Id. SinceΦ t ( x ) = − t tan α t tan α x + t (2 (cid:96) + tan α )1 + t tan α , if x ∈ [ a , a ];2 t tan α − t tan α x + t (2 (cid:96) − tan α )1 − t tan α , if x ∈ [ a , a + 1];Φ t ( x + k ) , if x + k ∈ [ a , a + 1] with k ∈ Z , (3.13)0 Lenci, Bonanno, Cristadoro then (cid:107) Φ t − Φ t (cid:107) ∞ ≤ (cid:96) + tan α ) (cid:18) α (cid:96) (cid:19) | t − t | . (3.14)We conclude by (3.12) and (3.14) that ρ is a Lipschitz function of t .Lastly, for a Lipshitz devil’s staircase the derivative exists almost everywhereand the following formula holds: (cid:90) t M ρ (cid:48) ( t ) dt = lim t → t − M ρ ( t ) − lim t → + ρ ( t ) = 1 . (3.15)Thus, m ( { t ∈ (0 , t M ) : ρ (cid:48) ( f t ) > } ) > . (3.16)On the other hand, by Proposition 3.4, ρ − ( Q \ Q o ) is a countable set and thus hasnull measure, while for all t ∈ ρ − ( Q o ), either ρ (cid:48) ( t ) is zero or it does not exist. Sincethe latter case amounts to a null-measure set in t , we have ρ − ( Q ) ⊆ { t ∈ (0 , t M ) : ρ (cid:48) ( f t ) = 0 } mod m, (3.17)where mod m means up to set of null m -measure. Passing to complements and using(3.16) yields the final assertion of the theorem. Q.E.D.
We give some final comments on the significance of Theorem 3.2. The set ofall rectangular trapezoids has 3 degrees of freedom, so the set of all internal-wavebilliard flows with unit speed in a rectangular trapezoid has 4 degrees of freedom.On the other hand, as for all billiard dynamics, a rescaling of the table gives riseto the same flow up to a rescaling of time. Moreover, an internal-wave billiard hasanother symmetry: a vertical or horizontal dilation of the table also leads to thesame flow up to a rescaling of time. So the effective degrees of freedom in thisproblem are 2.We can choose to fix (cid:96) > α, θ ). Theorem 3.2 proves two facts, the first of which already well-observednumerically, the second of which new, as far as we know:1. There exist Arnol’d tongues. An
Arnol’d tongue is the set of parametersfor which the rotation number of the map assumes a given rational value, provided it has positive measure in parameter space. Theorem 3.2 impliesthat all rotation numbers with odd denominator (in simplest terms) give riseto a tongue in the ( α, θ ) plane and these are the only ones occurring. Thiswas already observed in [14]. The phenomenon whereby a perturbation inparameter space does not change rotation number is known in the field as phase locking .2. The set of parameters ( α, θ ) for which all orbits of f , equivalently, all trajec-tories of the flow, are dense is —in physicist’s jargon — a fat fractal . More nternal-wave billiards t , equivalently, no interval in the parameter θ , may correspond onlyto irrational rotation numbers. Remark 3.5
The trapezoidal billiards considered in [14] depend on two parameters, d and τ , where τ > d ∈ ( − ,
1) is such that the shorter base is 1 + d (in our case it is (cid:96) ). In keepingwith the fact that two parameters are enough to describe all internal-billiard flows,Maas et al chose to fix θ = π/
4. The return map ˜ f they use is then defined on˜Σ = [ − τ, τ ). It follows that ˜ f = φ ◦ f ◦ φ − , where φ : [ − / , / −→ [ − τ, τ ) isgiven by φ ( x ) = 2 τ x and f is the map of Section 3 with parameters (cid:96) = 1 + d, α = arctan(2(1 − d )) , θ = arctan(2 τ ) . All results we prove for f hold for ˜ f with the corresponding parameters. In Fig. 6we show some of Arnol’d tongues in the ( d, τ )-plane computed rigorously with ourmethods. More in detail, the tongue for the rotation number p/q ( p, q positive andcoprime) starts when a is on a periodic orbit with rotation number p/q and endswhen a is on a periodic orbit of the same type. By Proposition 3.1, these twoconditions are different for q odd and coincident for q even, entailing phase lockingin the former case and no phase locking in the latter case. References [1]
C. Brouzet, E. Ermanyuk, S. Joubaud, G. Pillet and T. Dauxois , Internal waveattractors: different scenarios of instability , J. Fluid Mech. (2017), 544–568.[2]
C. Brouzet, I. N. Sibgatullin, H. Scolan, E. V. Ermanyuk and T. Dauxois , Internalwave attractors examined using laboratory experiments and 3D numerical simulations , J. FluidMech. (2016), 109–131.[3]
Y. Colin de Verdi`ere and L. Saint-Raymond , Attractors for two-dimensional waveswith homogeneous Hamiltonians of degree 0 , Comm. Pure Appl. Math. (2020), no. 2,421–462.[4] M. Degli Esposti, G. Del Magno and M. Lenci , Escape orbits and ergodicity in infinitestep billiards , Nonlinearity (2000), no. 4, 1275–1292.[5] W. de Melo and S. van Strien , One-dimensional dynamics , Ergebnisse der Mathematikund ihrer Grenzgebiete (3), 25. Springer-Verlag, Berlin, 1993.[6]
N. Grisouard, C. Staquet and I. Pairaud , Numerical simulation of a two-dimensionalinternal wave attractor , J. Fluid Mech. (2008), 1–14.[7]
J. Hazewinkel, N. Grisouard and S. B. Dalziel , Comparison of laboratory and numer-ically observed scalar fields of an internal wave attractor , Eur. J. Mech. B Fluids (2011),51–56. Lenci, Bonanno, Cristadoro
Figure 6:
Arnol’d tongues in the ( d, τ )-plane for some rotation numbers p/q . For q odd the figure shows only the boundaries of the tongue, whichis the set between the two curves of the same color, labeled ( p, q ). For q even the tongue reduces to a single curve. Compare with Fig. 2 of [14]. [8] J. Hazewinkel, C. Tsimitri, L. R. M. Maas and S. B. Dalziel , Observations on therobustness of internal wave attractors to perturbations , Phys. Fluids (2010), 107102, 9 pp.[9] J. Hazewinkel, P. van Breevoort, S. Dalziel and L. R. M. Maas , Observations onthe wavenumber spectrum and evolution of an internal wave attractor , J. Fluid Mech. (2008), 373–382.[10]
M.-R. Herman , Sur la conjugaison diff´erentiable des diff´eomorphismes du cercle `a des rota-tions , Inst. Hautes ´Etudes Sci. Publ. Math. (1979), 5–233.[11] A. Katok and B. Hasselblatt , Introduction to the modern theory of dynamical systems ,Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cam-bridge, 1995.[12]
F.-P. A. Lam and L. R. M. Maas , Internal wave focusing revisited; a reanalysis and newtheoretical links , Fluid Dynam. Res. (2008), no. 2, 95–122.[13] L. R. M. Maas , Wave attractors: linear yet nonlinear , Internat. J. Bifur. Chaos Appl. Sci.Engrg. (2005), no. 9, 2557–2782.[14] L. R. M. Maas, D. Benielli, J. Sommeria and F.-P. A. Lam , Observation of an internalwave attractor in a confined, stably stratified fluid , Nature (1997), 557–561.[15]
L. R. M. Maas and F.-P. A. Lam , Geometric focusing of internal waves , J. Fluid Mech. (1995), 1–41.[16]
A. M. M. Manders, J. J. Duistermaat and L. R. M. Maas , Wave attractors in asmooth convex enclosed geometry , Phys. D (2003), no. 3-4, 109–132. nternal-wave billiards [17] G. Pillet, L. R. M. Maas and T. Dauxois , Internal wave attractors in 3D geometries:a dynamical systems approach , Eur. J. Mech. B Fluids (2019), 1–16.[18] B. R. Sutherland , Internal gravity waves , Cambridge University Press, Cambridge, 2010.[19]