A Classification of braid types for periodic orbits of diffeomorphisms of surfaces of genus one with topological entropy zero

Publicacions Matemátiques, Vol 35 (1991), 543-558 . A CLASSIFICATION OF BRAID TYPES FOR PERIODIC ORBITS OF DIFFEOMORPHISMS OF SURFACES OF GENUS ONE WITH TOPOLOGICAL ENTROPY ZERO Abstract GUASCHI, J . LLIBRE AND R .S . MACKAY We classify the braid types that can occur for finite unions of periodic orbits of diffeomorphisms of surfaces of genus one with zero topological entropy . 1 . Introduction In this paper we classify the braid types that can occur for finite unions of periodic orbits of diffeomorphisms of surfaces of genus one with zero topological entropy . This extends the analysis from the case of genus zero [LMI] . The case of most interest to us is diffeomorphisms of the torus, isotopic to the identity . This is relevant to the behaviour of three coupled oscillators, for example . A good picture of their dynamics is developing [KMG], [LM2],[MZ], [H2], [F], [BGKM] . We hope that our results will helo solve the intriguing problem of understanding the boundary of zero topological entropy in the space of C 1 diffeomorphisms of the torus . We begin by establishing some notation and recalling the definition of braid type . Let f : X -> X be a diffeomorphism of an oriented manifold X . Write h(f) for the topological entropy of f . We write o - p, o - r for orientationpreserving and reversing, respectively . Given two diffeomorphisms f : X - ; X, g : Y ~ Y of oriented manifolds, we write f - g if there exists an conjugacy between them . Let M be a súrface, i .e . a compact connected oriented f : M -> M be a diffeomorphism of M and let P be a finite orbits for f . Then we define f-p : Mp , M-p by removing P from M and recompactifying by replacing the points of P by circles on which fP is the projective action of D f [B] . o-p 2-manifold . Let union of periodic 54 4  J . GUASCHI, J . LLIBRE, R .S . MACKAY Given two diffeomorphisms f, g : X -> X of a surface X and finite unions P, Q of periodic orbits for f, g respectively, we say the pairs (P, f) and (2, g) have the same braid type if there exists an op homeomorphism k : XP - XQ such that k fP k -1 is isotopic to gQ . The equivalence class of (P, f ), denoted [P, f ], is called its braid type . To specify the braid type of a pair (P, f) we will use Nielsen-Thurston theory to select a simplest representative of the equivalence class . We refer the reader to [LM1] for the necessary information about Nielsen-Thurston theory of classification of surface homeomorphisms up to isotopy, and the definitions of the classes of diffeomorphisms which we call dise trees, reversing disc trees and reversing annulus trees . The plan of the paper is as follows . In Section 2, we classify finite order homeomorphisms of the torus, which is an important preliminary result . Then in Section 3, we use Nielsen-Thurston theory to isotopc fP to a standard form, and analyse the possibilities . Our main result is Theorem 4, but because it is rather long to state and requires notions introduced in Section 2, we leave its statement until Section 3 . In Section 4 we rederive results of [H1], [H2], and [LM2] as corollaries of those of Section 3 . We thank the referee for a careful reading . The first two authors were partially supported by a SERC studentship and a DGICYT grant respectively . 2 . Classification of finite order homeomorphisms of the torus In this section, we give a classification of finite order homeomorphisms of the torus up to o - p conjugacy . Let T x , x E R2, be translation by x on R2, Le . T x (y) = y + x, yE ff8 2 . Let R 4  w E R/27r7L be rotation about 0 on R2 by angle w . Let rbe the reflection {x / = x r y , _ -y on IE8 2 .  Define ro to be the group generated by Ti l o l and Tl o , l l, Fo that generated by T(I,o) and T(2, ~) . If f : R2 _ R2 commutes with the action of a group I' on R2, we define f/I' : R 2 /I' , R 2 /I' by identifying points of the same orbit under I' . For q E NN, let wbe the quotient of Z under the equivalence relation generated by the relations p - p + q, p - -p . Theorem 1 . If f : T 2 -> T 2 is a homeomorphism of finite order, then it is o - p conjugate to one of the following : BRAID TYPES WITH ZERO ENTROPY  54 5 (a) T(. oyq/I'o for some q E NI, p E íq, with p, q having no common factor (order q) . (b) R u ,/Fo, w= for/2, w, or R W /Fá, w= f7r/3, f27r/3 (orders 4,,2,6,,1 respectively) (see Figure 1) . (c) ro T( p o)1 q /Fo, for some q E N1, p E  with p, qhaving no common factor (order q if q is even, order 2q if q is odd) . (d) r o R, r /2/Fo (order 2) (see Figure 2) . No two of the aboye are o - p conjugate . The proof of Theorem 1 reduces to the classification of isometries of tori, a well-studied problem (e .g . see [NS]) . However, we have not found a compact treatment in the literature, nor one which considers the question of which isometries are equivalent up to o - p conjugacy . So we give a derivation in the Appendix . For case (b) of Theorem 1, the numbers of periodie orbits of smaller period than the order are given by the following theorem . Theorem 2 ([E]) . The rotation of order 2 has 4 fixed points . The rotations of order 3 have 3 fixed points . The rotations of order 4 have 2 fixed points and one orbit of period 2 . The rotations of order 6 have one fixed point, one orbit of period 2, and one orbit of period 3 . (See Figure 1) . 3 . Results We firstly recall some definitions and theorems that we will need . We then go on to state and prove our results . Let f : M -> M be a homeomorphism of a surface of genus one, then its completion g : T 2 _ T 2 , is defined by considering M to be T 2 minus a disjoint union of equal size dises D i , and extending glaD ; radially into D i [E] . If h(f) = 0, then h(g) = 0 . We say a simple closed curve on M is rotational if it is homotopically non-trivial on T 2 after filling in the holes . We recall the following two results . Theorem A ([LMl]) . Let f : X -> X be a homeomorphism of a surface of genus zero . If f is o - p, it has either a fixed point or an invariant boundary component . If f is o - r it has either a fixed point or an invariant boundary component, or an orbit of period 2 or a boundary component of period 2 . So in both the o-p and o-r cases, if P is a finite union of periodic orbits for f, and fP does not have a boundary component of period 1 (o - p) or period 1 or 2 (o - r), we can always append a fixed point or period 2 orbit to P in order to achieve this . 54 6  J . GUASCHI, J . LLIBRE, R .S . MACKAY Theorem B ([LMl]) . Let f : X --> X be a diffeomorphism of a surface of genus zero, with h(f) = 0, and let P be a finite union of periodic orbits . (1) If P contains a fixed point or f has an invariant boundary component, then (a) If f is o - p, [P, f ] has a representative which is a dise tree . (b) If f is o - r, [P, f] has a representative which is a reversing dise tree . (2) If f is o - r, and P contains no fixed point and f has no invariant boundary component, but either P has a point of period ,2 or f has a boundary component of period 2, then [P, f] has a representative which is a reversing annulus Cree . Let f : M , M be a diffeomorphism of a surface of genus one with h(f) = 0, and let P be a finite union of periodic orbits for f . Let F be a Thurston canonical form for fP . Then from Thurston's classification of surface homeomorphisms, F is either reducible, or of finite order . First we consider the case that F has a rotational reducing curve . Theorem 3 . Suppose f : M --> M is a diffemorphism of a surface of genus one with h(f) = 0, and P is a finite union of periodic orbits for f . Let F be a Thurston canonical form for fv, such that F has a rotational reducing curve F, of period p . Remove the tubular neighbourhood of F and its images, to obtain a disjoint union of punctured annuli A i . Then (i) If all Aá have period p, then [P, f ] has a representative which is p annuli, joined by generalised twists, permuted like a rotation, and if f is o - p or p is even there exists a dise tree d : X' ~ X' such that FPI A j - d or if f is o - r and p is odd there exists a reversing dise tree d' : X' -+ X' such that FPIAj - d' . (ii) Suppose some Ai has period q :~ p (so q = p/2), then p = 2, and there are two such annuli Al, A 2 . If f is o - p, [P, f] has a representative which is two annuli, joined by generalised twists, and there exist dise trees Di : X' -+ X', i = 1, 2 such that FI Ai, - D i .  If f is or, [P, f ] has a representative which is two annuli, joined by generalised twists, and for each of Al, A 2 there exist reversing dise trees D' : X' , X' such that FIAj - D' or reversing annulus trees S i : X --> X, such that FIAj - Si, according as FIAj has an invariant boundary component or fixed point, or not . Proof . Suppose F has period p . Define A = {Fkr : 0 < k < p} . Then F permutes the elements of A . If we remove the tubular neighbourhood of F and its images, we obtain a disjoint union of annuli A i with holes, which are BRAID TYPES WITH ZERO ENTROPY  547 permuted . Then the Ai either have period p, or if p is even, some of the A i may have period p/2 (Le . the two boundaries of A i in A are interchanged by Fp/ 2 ) . This gives two cases : (i) If the A i have period p, F permutes the Ai, and there are two subcases : (a) If f is o - p, or if f is o - r and p is even, then FP I A ¡ is o - p, so by Theorem B, there exists a disc tree d : X' , X' such that FP I A j - d . (b) If f is o - r and p is odd, then FPI Aj is o - r, so by Theorem B, there exists a reversing disc tree d' : X' - X' such that FPIAj - d' . (ii) Suppose p is even, and there exists some A i (without loss of generality i = 0) such that Ai has period p/2 . Put q = p/2 . Then we claim that q=1 . To prove the claim, suppose Ao has period q > 1 . Write Ak = Fk(Ao), 1 < k < q . Consider the situation on the torus ; then each Ak has two boundary components rk ll , rk 2 ) E A, say, for 0 <_ k < q . Then for j =~ k, I'~') ~ rkl, i l = 1, 2, Le . n o two distinct Ak have a boundary component in common . For suppose FjZI . = rk ¿ l for some i, l E {1, 2}, j =,A k (see Figure 3(a)) . Then since F 9 interchanges rk l l and rk 2 ) for each k, then F 9 (r~ l l) = Fui--1) = F9(Fkil) Fkt+I) . hence there are only two distinct elements of A, so q = 1, a contradiction . Since the Ak have no boundary components in common, there exist precisely q tubular regions Bl, . . . , B 9 , such that each Bi lies between two Ak, Le . each Bi has two distinct elements of A as its (rotational) boundary components . Consider one such element of {Bi}9 1 , B, say . Then without loss of generality, it lies between Ao and Ak, for some k, and has boundary components róll, 1'kl) E A (see Figure 3(b)) . B is invariant under F 2q . Since r has period 2q, B must have period either q or 2q . Suppose it has period q, then F 9 interchanges its boundary components, soFk l) = F 9 (ro l » . But F 9 interchanges the boundary components of Ao so F 9 (Fó ') ) =ro e ) , therefore Ao and Ak have a boundary component in common, which we have shown does not occur . Hence B has period 2q . However F k (Ao) = Ak, fór k < q, and considering F k +q(Ao) = Ak, if necessary, then F k (r( 1 » = Fkl), and F k (ro 21 ) = rk 21 . But the elements of A are permuted like a rotation or rotation with reflection by F, since f is a diffeomorphism, hence F k (Fk l » = ról), therefore F k (B) = B, a contradiction, as B has period 2q . This proves the claim . So q = 1, and r has period 2, with A = {F, F(F)} . If we remove the tubular neighbourhoods of r and F(I'), we obtain two disjoint annuli A o and A1 . Then there are two subcases to consider : 54 8  J . GUASCHI, J . LLIBRE, R .S . MACKAY (a) If f is o - p, then FIA j (i = 0, 1) is o - p, so by Theorem B, [P, f] has a representative which is two annuli joined by generalised twists, such that FI A j is a dise tree . (b) If f is r - o, then FEA ¡ (i = 0,1) is o - r, so by Theorem B, ['P, f] has a representative which is two annuli joined by generalised twists, such that FI A j is a reversing dise tree or a reversing annulus tree, according as f has an invariant boundary component or P contains a fixed point in Ai, or not . This completes the proof of Theorem 3 . Next we consider the case where there is no rotational reducing curve . Either F is of finite order, or has a non-rotational reducing curve C, in which case we may remove the decomposition componente (of genus zero) of C and its images from M . So in both cases, we can find a unique decomposition component S of genus one, such that FAS is of finite order . Let G : T2 -> T 2 be the completion of FI S . Then G is of finite order, so is o - p conjugate to one of the cases of Theorem 1 . This leads to the following theorem, which is our main result : Theorem 4 . Suppose F : M - M is a Thurston canonical form for a homeomorphism of a surface of genus one, with unique decomposition component S of genus one . Let G : T 2 -> T 2 be the completion of FAS . Then one of the following is true : (a) GT(P .o)/q/ro, and all boundary componente of S have period q . F~arther, for each orbit a l ,. .., á q of boundary componente of S not in aM, there exists a dise tree d : X' , X' such that the component Xi of M\S inside Ói is homeomorphic to X', and F 9 IXi - d . (b) G - R,/I'o, w = f7r/2, 7r (order k = 4, 2), or GR  ,/ro, w = for/3, ±27x/3 (order k = 6, 3) . If k = 2, then all boundary components of S have period 1 or 2, with at most 4 of period 1 .  If k = 3, then all boundary components of S Nave period 1 or 3, with at most 3 of period 1 . If k = 4, then all boundary components of S have period 1, 2 or 4, with at most 2 of period 1, and 1 orbit of period 2 . If k = 6, then all boundary components of S have period 1, 2, 3 or 6 with at most 1 of period 1, 1 orbit of period 2, and 1 orbit of period 3 . In case, for each orbit of boundary components of period p, al, . . . , O P , of S, not in OM, there exists a dise tree d : X' -4 X' such that the component X i of M\S inside á i is homeomorphic to X', and FPIXi - d . (e) (i) G- (r o TOP o)/q)/Po .  If q is even, all boundary components have period q . If q is odd, all boundary components have period q or 2q . (ii) G- (roR,/2)/I'o . All boundary components of S have period either 1 or 2 . BRAID TYPES WITH ZERO ENTROPY  54 9 In both parts of case (c), for each orbit of boundary components of period p, 91 5 . . . , ap, of S not in OM, there exists a disc tree d : X' - X' such that the component Xi of M\S inside Vi is homeomorphic to X' and F P I Xi - d if p is even, or there exists a reversing disc tree d' : X' - X' such that the component Xi of M\S inside al is homeomorphic to X' and F P I Xi -d if p is odd . Proof .. We apply Theorem 1 to the completion G : T2 , T 2 of FAS . Case (a) G ^- TO P o)/q/I'o, so all points of T 2 have period q under G . Hence all boundary components of S have period q, and if we consider the orbits a l , . . . . 0, of those not in aM, the corresponding decomposition components Xi of M\S inside al are of genus zero, and we may apply Theorem B, so there exists a disc tree d : X' -> X' such that X i is homeomorphic to X', and F 4 1Xi - d . Case (b) G - R  ,/I'o, w= f7r/2, n : k = 4, 2, or G - Ru,/I'o, w= t7r/3, ±27r/3 : k = 6, 3 . The statements about the orbits of boundary components are an immediate consequence of applying Theorem 2 to the completion G : T 2 -> T 2 . As in case (a), since G is op, we obtain disc trees in each orbit of boundary components of S not in 8M . Case (c) (i) G- (r oTO P o)/ a )/I'o . From the proof of Theorem 1, the boundary components of S have period q if q is even, or they have period q or 2q if q is odd . Case (c) (ü) G - (r o R,r/2)/I'o . Again from the proof of Theorem 1, the boundary components of S have period 1 or 2 . In both parts of case (c), each component of T 2 \S inside a boundary component not in 8M of even period r contains a disc tree since Fr is o - p, whilst those of odd period r' contain a reversing disc tree since Fr' is o-r, by Theorem B . 55 0  J . GUASCHI, J . LLIBRE, R .S . MACKAY 4 . Two Corollaries As a corollary of the results of Section 3 we obtain the genus one case of [H1] : Theorem 5 . Let f : M -M be an ordiffeomorphism of a surface M of genus one . If f has periodic orbits, or orbits of boundary components, with 3 distinct odd peros, then h(f) > 0 . To derive this from the above, we require the genus zero result of [BF], [H1], also derived by [LM1] : Theorem C . Let f : X-X be an o - r diffeomorphism of a surface of genus zero . If f has periodic orbits or orbits of boundary components with two distinct odd peros, then h(f) > 0 . Proof of Theorem 5 : Let f : M -~ M be an or diffeomorphism of a surface M of genus one . Let P be the union of three orbits of distinct odd period . Let F be a Thurston canonical form for fP . Suppose h(f) -- 0 . Then there are two cases : Case (a) : Suppose there is a rotational reducing curve, I', say, with period p . Remove its annular neighbourhood and its images, then we obtain a disjoint union of annuli A i . From Theorem 3, there are two possibilities . In the first case, the A i have period p, so the Ai are permuted by F, and thus p divides the order of each periodic orbit, hence p is odd . So if we consider any Ai, F P ¡A¡ is o - r, since f is o - r . Ai must contain boundary components corresponding to the three orbits of distinct odd period . So by Theorem C, h(f) > 0, a contradiction . The other possibility is when p = 2, and there are two invariant annuli Ao, Al . Then one of Ao, Al must contain boundary components corresponding to at least two of the three orbits, and since FIA jj (i = 0,1) is o - r, Theorem C implies that h(f) > 0, a contradiction . Case (b) : Suppose there is no rotational reducing curve . Then there exists a unique decomposition component S of genus one . Let G : T2 -> T2 be the completion of FAS . Thenwe are in Case (c) of Theorem 4 . If G- (r-T(P o)M/I'o, all boundary components of S llave period q or 2q . If G- (r o R,/2)/I'o, then all boundary components of S llave period 1 or 2 . In particular, all boundary components of S of odd period llave the same period p, and the remaining boundary components corresponding to P must lie within decomposition components X i of genus zero whose outer boundaries have period p . Since F P I Xi is o-r, and at least one of the X i contains boundary components of odd order, BRAID TYPES WITH ZERO ENTROPY  55 1 not p, corresponding to the orbits of P, then Theorem C implies that h (f ) > 0, a contradiction . As a second corollary we will derive a result of [LM2], [H2] for diffeomorphisms of the torus isotopic to the identity . We refer the reader to [LM2] for definitions of lifts, rotation vectors, etc . We recall that for a continuous map f : T2 , T2 with lift f : R2 , If8 2 , if I is the group of integer translations y ,, : x 1--> x + m, xE R2, m E Z2, and if f is homotopic to the identity, then fy = -y f for all " y EF . Theorem 6 . Let f : T 2 - 3 T 2 be a homeomorphism of the torus isotopic to the identity, an .d suppose h(f) = 0 . Then all rotation vectors associated with the periodic orbits of f are collinear . Proof .. Suppose that f has a finite union of periodic orbits, with associated distinct rotation vectors pi/gi, i = 1, . . . , N . Then from [LM2], since f is homotopic to the identity, for each i E {1, . . . , N} there exists a periodic orbit Qá of primitive rotation type (pi, qi) . Let P =UN 1 Qá, and let F be a Thurston canonical form for fP . Suppose F is of finite order, then using (*) and Theorem 1, we see that F - T(P o)1e/I'o, for some q E N, pE w, so all points of T 2 are periodic with period qand rotation vector (p, 0)/q . The remaining possibility is that F is reducible (with finite order components, though we shall not need this) . There are no non-rotational reducing curves for F . For suppose F were such a curve, then it must surround at least two holes, but they must come from the same orbit, so the rotation type of that orbit cannot be primitive, which is a contradiction . Suppose there is a rotational reducing curve 1' for F . Let G : T 2 -> T 2 be the completion of F, and let G : R2 --> R2 be a lift of G . Let m E 71 2 \{0} be the homotopy type of 1', and q be its period . Then F lifts to an infinite set of curves F, each invariant under T  ,,, which partition the plane hito infinite strips S each within a bounded distance of some straight line of direction m . Furthermore, there exists pE 71 2 such that FqT F = 1', and since F is invertible the same holds for the strips S . Hence the rotation set of F, and in particular the rotation vectors of the chosen periodic orbits, are contained in the straight line {p/q + tm : t E IR} . This completes the proof . Appendix : Proof of Theorem 1 To prove Theorem 1, we require the following : 558  J . GUASCHI, J . LLIBRE, R .S . MACKAY [NS] V .V . NIKULIN AND I .R . SHAFAREVICH, "Geometries and groups," Springer, 1987 . [S] R .L.E . SCHWARZENBERGER, "N dimensional crystallography," Pitman research notes in Mathematics 41, 1980 . [W] J.A . WOLF, "Spaces of constant curvature," McGraw-Hill, 1967 . J . Guaschi : Nonlinear Systems Laboratory Mathematics Institute University of Warwick Coventry CV4 7AL ENGLAND J . Llibre : Departament de Matemátiques Facultat de Ciéncies Universitat Autónoma de Barcelona 08193 Bellaterra (Barcelona) SPAIN R .S . Mackay : Nonlinear Systems Laboratory Mathematics Institute University of Warwick Coventry CV47AL ENGLAND Primera versió rebuda el 23 d'Agost de 1990, darrera versió rebuda el 2 de Setembre de 1991