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

Publicacions Ma emá iques, Vol 35 (1991), 543-558 . A CLASSIFICATION OF BRAID TYPES FOR PERIODIC ORBITS OF DIFFEOMORPHISMS OF SURFACES OF GENUS ONE WITH TOPOLOGICAL ENTROPY ZERO Abs ac GUASCHI, J . LLIBRE AND R .S . MACKAY We classi y he b aid ypes ha can occu o ini e unions o pe iodic o bi s o di eomo phisms o su aces o genus one wi h ze o opological en opy . 1 . In oduc ion In his pape we classi y he b aid ypes ha can occu o ini e unions o pe iodic o bi s o di eomo phisms o su aces o genus one wi h ze o opological en opy . This ex ends he analysis om he case o genus ze o [LMI] . The case o mos in e es o us is di eomo phisms o he o us, iso opic o he iden i y . This is ele an o he beha iou o h ee coupled oscilla o s, o example . A good pic u e o hei dynamics is de eloping [KMG], [LM2],[MZ], [H2], [F], [BGKM] . We hope ha ou esul s will helo sol e he in iguing p oblem o unde s anding he bounda y o ze o opological en opy in he space o C 1 di eomo phisms o he o us . We begin by es ablishing some no a ion and ecalling he de ini ion o b aid ype . Le : X -> X be a di eomo phism o an o ien ed mani old X . W i e h( ) o he opological en opy o . We w i e o - p, o - o o ien a ion- p ese ing and e e sing, espec i ely . Gi en wo di eomo phisms : X - ; X, g : Y ~ Y o o ien ed mani olds, we w i e - g i he e exis s an conjugacy be ween hem . Le M be a sú ace, i .e . a compac connec ed o ien ed : M -> M be a di eomo phism o M and le P be a ini e o bi s o . Then we de ine -p : Mp , M-p by emo ing P om M and ecompac i ying by eplacing he poin s o P by ci cles on which P is he p ojec i e ac ion o D [B] . o-p 2-mani old . Le union o pe iodic 54 4  J . GUASCHI, J . LLIBRE, R .S . MACKAY Gi en wo di eomo phisms , g : X -> X o a su ace X and ini e unions P, Q o pe iodic o bi s o , g espec i ely, we say he pai s (P, ) and (2, g) ha e he same b aid ype i he e exis s an o- p homeomo phism k : XP - XQ such ha k P k -1 is iso opic o gQ . The equi alence class o (P, ), deno ed [P, ], is called i s b aid ype . To speci y he b aid ype o a pai (P, ) we will use Nielsen-Thu s on he- o y o selec a simples ep esen a i e o he equi alence class . We e e he eade o [LM1] o he necessa y in o ma ion abou Nielsen-Thu s on heo y o classi ica ion o su ace homeomo phisms up o iso opy, and he de ini ions o he classes o di eomo phisms which we call dise ees, e e sing disc ees and e e sing annulus ees . The plan o he pape is as ollows . In Sec ion 2, we classi y ini e o de homeomo phisms o he o us, which is an impo an p elimina y esul . Then in Sec ion 3, we use Nielsen-Thu s on heo y o iso opc P o a s anda d o m, and analyse he possibili ies . Ou main esul is Theo em 4, bu because i is a he long o s a e and equi es no ions in oduced in Sec ion 2, we lea e i s s a emen un il Sec ion 3 . In Sec ion 4 we ede i e esul s o [H1], [H2], and [LM2] as co olla ies o hose o Sec ion 3 . We hank he e e ee o a ca e ul eading . The i s wo au ho s we e pa - ially suppo ed by a SERC s uden ship and a DGICYT g an espec i ely . 2 . Classi ica ion o ini e o de homeomo phisms o he o us In his sec ion, we gi e a classi ica ion o ini e o de homeomo phisms o he o us up o o - p conjugacy . Le T x , x E R2, be ansla ion by x on R2, Le . T x (y) = y + x, yE 8 2 . Le R 4  w E R/27 7L be o a ion abou 0 on R2 by angle w . Le be he e lec ion {x / = x y , _ -y on IE8 2 .  De ine o o be he g oup gene a ed by Ti l o l and Tl o , l l, Fo ha gene a ed by T(I,o) and T(2, ~) . I : R2 _ R2 commu es wi h he ac ion o a g oup I' on R2, we de ine /I' : R 2 /I' , R 2 /I' by iden i ying poin s o he same o bi unde I' . Fo q E NN, le wbe he quo ien o Z unde he equi alence ela ion gene a ed by he ela ions p - p + q, p - -p . Theo em 1 . I : T 2 -> T 2 is a homeomo phism o ini e o de , hen i is o - p conjuga e o one o he ollowing : BRAID TYPES WITH ZERO ENTROPY  54 5 (a) T(. oyq/I'o o some q E NI, p E íq, wi h p, q ha ing no common ac o (o de q) . (b) R u ,/Fo, w= o /2, w, o R W /Fá, w= 7 /3, 27 /3 (o de s 4,,2,6,,1 espec i ely) (see Figu e 1) . (c) o T( p o)1 q /Fo, o some q E N1, p E  wi h p, qha ing no common ac o (o de q i q is e en, o de 2q i q is odd) . (d) o R, /2/Fo (o de 2) (see Figu e 2) . No wo o he aboye a e o - p conjuga e . The p oo o Theo em 1 educes o he classi ica ion o isome ies o o i, a well-s udied p oblem (e .g . see [NS]) . Howe e , we ha e no ound a compac ea men in he li e a u e, no one which conside s he ques ion o which isome ies a e equi alen up o o - p conjugacy . So we gi e a de i a ion in he Appendix . Fo case (b) o Theo em 1, he numbe s o pe iodie o bi s o smalle pe iod han he o de a e gi en by he ollowing heo em . Theo em 2 ([E]) . The o a ion o o de 2 has 4 ixed poin s . The o a ions o o de 3 ha e 3 ixed poin s . The o a ions o o de 4 ha e 2 ixed poin s and one o bi o pe iod 2 . The o a ions o o de 6 ha e one ixed poin , one o bi o pe iod 2, and one o bi o pe iod 3 . (See Figu e 1) . 3 . Resul s We i s ly ecall some de ini ions and heo ems ha we will need . We hen go on o s a e and p o e ou esul s . Le : M -> M be a homeomo phism o a su ace o genus one, hen i s comple ion g : T 2 _ T 2 , is de ined by conside ing M o be T 2 minus a disjoin union o equal size dises D i , and ex ending glaD ; adially in o D i [E] . I h( ) = 0, hen h(g) = 0 . We say a simple closed cu e on M is o a ional i i is homo opically non- i ial on T 2 a e illing in he holes . We ecall he ollowing wo esul s . Theo em A ([LMl]) . Le : X -> X be a homeomo phism o a su ace o genus ze o . I is o - p, i has ei he a ixed poin o an in a ian bounda y componen . I is o - i has ei he a ixed poin o an in a ian bounda y componen , o an o bi o pe iod 2 o a bounda y componen o pe iod 2 . So in bo h he o-p and o- cases, i P is a ini e union o pe iodic o bi s o , and P does no ha e a bounda y componen o pe iod 1 (o - p) o pe iod 1 o 2 (o - ), we can always append a ixed poin o pe iod 2 o bi o P in o de o achie e his . 54 6  J . GUASCHI, J . LLIBRE, R .S . MACKAY Theo em B ([LMl]) . Le : X --> X be a di eomo phism o a su ace o genus ze o, wi h h( ) = 0, and le P be a ini e union o pe iodic o bi s . (1) I P con ains a ixed poin o has an in a ian bounda y componen , hen (a) I is o - p, [P, ] has a ep esen a i e which is a dise ee . (b) I is o - , [P, ] has a ep esen a i e which is a e e sing dise ee . (2) I is o - , and P con ains no ixed poin and has no in a ian bounda y componen , bu ei he P has a poin o pe iod ,2 o has a bounda y componen o pe iod 2, hen [P, ] has a ep esen a i e which is a e e sing annulus C ee . Le : M , M be a di eomo phism o a su ace o genus one wi h h( ) = 0, and le P be a ini e union o pe iodic o bi s o . Le F be a Thu s on canonical o m o P . Then om Thu s on's classi ica ion o su ace homeo- mo phisms, F is ei he educible, o o ini e o de . Fi s we conside he case ha F has a o a ional educing cu e . Theo em 3 . Suppose : M --> M is a di emo phism o a su ace o genus one wi h h( ) = 0, and P is a ini e union o pe iodic o bi s o . Le F be a Thu s on canonical o m o , such ha F has a o a ional educing cu e F, o pe iod p . Remo e he ubula neighbou hood o F and i s images, o ob ain a disjoin union o punc u ed annuli A i . Then (i) I all Aá ha e pe iod p, hen [P, ] has a ep esen a i e which is p annuli, joined by gene alised wis s, pe mu ed like a o a ion, and i is o - p o p is e en he e exis s a dise ee d : X' ~ X' such ha FPI A j - d o i is o - and p is odd he e exis s a e e sing dise ee d' : X' -+ X' such ha FPIAj - d' . (ii) Suppose some Ai has pe iod q :~ p (so q = p/2), hen p = 2, and he e a e wo such annuli Al, A 2 . I is o - p, [P, ] has a ep esen a i e which is wo annuli, joined by gene alised wis s, and he e exis dise ees Di : X' -+ X', i = 1, 2 such ha FI Ai, - D i .  I is o- , [P, ] has a ep esen a i e which is wo annuli, joined by gene alised wis s, and o each o Al, A 2 he e exis e e sing dise ees D' : X' , X' such ha FIAj - D' o e e sing annulus ees S i : X --> X, such ha FIAj - Si, acco ding as FIAj has an in a ian bounda y componen o ixed poin , o no . P oo . Suppose F has pe iod p . De ine A = {Fk : 0 < k < p} . Then F pe mu es he elemen s o A . I we emo e he ubula neighbou hood o F and i s images, we ob ain a disjoin union o annuli A i wi h holes, which a e BRAID TYPES WITH ZERO ENTROPY  547 pe mu ed . Then he Ai ei he ha e pe iod p, o i p is e en, some o he A i may ha e pe iod p/2 (Le . he wo bounda ies o A i in A a e in e changed by Fp/ 2 ) . This gi es wo cases : (i) I he A i ha e pe iod p, F pe mu es he Ai, and he e a e wo subcases : (a) I is o - p, o i is o - and p is e en, hen FP I A ¡ is o - p, so by Theo em B, he e exis s a disc ee d : X' , X' such ha FP I A j - d . (b) I is o - and p is odd, hen FPI Aj is o - , so by Theo em B, he e exis s a e e sing disc ee d' : X' - X' such ha FPIAj - d' . (ii) Suppose p is e en, and he e exis s some A i (wi hou loss o gene ali y i = 0) such ha Ai has pe iod p/2 . Pu q = p/2 . Then we claim ha q=1 . To p o e he claim, suppose Ao has pe iod q > 1 . W i e Ak = Fk(Ao), 1 < k < q . Conside he si ua ion on he o us ; hen each Ak has wo bounda y componen s k ll , k 2 ) E A, say, o 0 <_ k < q . Then o j =~ k, I'~') ~ kl, i l = 1, 2, Le . n o wo dis inc Ak ha e a bounda y componen in common . Fo suppose FjZI . = k ¿ l o some i, l E {1, 2}, j =,A k (see Figu e 3(a)) . Then since F 9 in e changes k l l and k 2 ) o each k, hen F 9 ( ~ l l) = Fui--1) = F9(Fkil) Fk +I) . hence he e a e only wo dis inc elemen s o A, so q = 1, a con adic ion . Since he Ak ha e no bounda y componen s in common, he e exis p ecisely q ubula egions Bl, . . . , B 9 , such ha each Bi lies be ween wo Ak, Le . each Bi has wo dis inc elemen s o A as i s ( o a ional) bounda y componen s . Conside one such elemen o {Bi}9 1 , B, say . Then wi hou loss o gene - ali y, i lies be ween Ao and Ak, o some k, and has bounda y componen s óll, 1'kl) E A (see Figu e 3(b)) . B is in a ian unde F 2q . Since has pe iod 2q, B mus ha e pe iod ei he q o 2q . Suppose i has pe iod q, hen F 9 in e - changes i s bounda y componen s, soFk l) = F 9 ( o l » . Bu F 9 in e changes he bounda y componen s o Ao so F 9 (Fó ') ) = o e ) , he e o e Ao and Ak ha e a bounda y componen in common, which we ha e shown does no occu . Hence B has pe iod 2q . Howe e F k (Ao) = Ak, ó k < q, and conside ing F k +q(Ao) = Ak, i nec- essa y, hen F k ( ( 1 » = Fkl), and F k ( o 21 ) = k 21 . Bu he elemen s o A a e pe mu ed like a o a ion o o a ion wi h e lec ion by F, since is a di eo- mo phism, hence F k (Fk l » = ól), he e o e F k (B) = B, a con adic ion, as B has pe iod 2q . This p o es he claim . So q = 1, and has pe iod 2, wi h A = {F, F(F)} . I we emo e he ubula neighbou hoods o and F(I'), we ob ain wo disjoin annuli A o and A1 . Then he e a e wo subcases o conside : 54 8  J . GUASCHI, J . LLIBRE, R .S . MACKAY (a) I is o - p, hen FIA j (i = 0, 1) is o - p, so by Theo em B, [P, ] has a ep esen a i e which is wo annuli joined by gene alised wis s, such ha FI A j is a dise ee . (b) I is - o, hen FEA ¡ (i = 0,1) is o - , so by Theo em B, ['P, ] has a ep esen a i e which is wo annuli joined by gene alised wis s, such ha FI A j is a e e sing dise ee o a e e sing annulus ee, acco ding as has an in a ian bounda y componen o P con ains a ixed poin in Ai, o no . This comple es he p oo o Theo em 3 . Nex we conside he case whe e he e is no o a ional educing cu e . Ei he F is o ini e o de , o has a non- o a ional educing cu e C, in which case we may emo e he decomposi ion componen e (o genus ze o) o C and i s images om M . So in bo h cases, we can ind a unique decomposi ion componen S o genus one, such ha FAS is o ini e o de . Le G : T2 -> T 2 be he comple ion o FI S . Then G is o ini e o de , so is o - p conjuga e o one o he cases o Theo em 1 . This leads o he ollowing heo em, which is ou main esul : Theo em 4 . Suppose F : M - M is a Thu s on canonical o m o a homeomo phism o a su ace o genus one, wi h unique decomposi ion compo- nen S o genus one . Le G : T 2 -> T 2 be he comple ion o FAS . Then one o he ollowing is ue : (a) G- T(P .o)/q/ o, and all bounda y componen e o S ha e pe iod q . F~a - he , o each o bi a l ,. .., á q o bounda y componen e o S no in aM, he e exis s a dise ee d : X' , X' such ha he componen Xi o M S inside Ói is homeomo phic o X', and F 9 IXi - d . (b) G - R,/I'o, w = 7 /2, 7 (o de k = 4, 2), o G- R  ,/ o, w = o /3, ±27x/3 (o de k = 6, 3) . I k = 2, hen all bounda y componen s o S ha e pe iod 1 o 2, wi h a mos 4 o pe iod 1 .  I k = 3, hen all bounda y componen s o S Na e pe iod 1 o 3, wi h a mos 3 o pe iod 1 . I k = 4, hen all bounda y componen s o S ha e pe iod 1, 2 o 4, wi h a mos 2 o pe iod 1, and 1 o bi o pe iod 2 . I k = 6, hen all bounda y componen s o S ha e pe iod 1, 2, 3 o 6 wi h a mos 1 o pe iod 1, 1 o bi o pe iod 2, and 1 o bi o pe iod 3 . In case, o each o bi o bounda y componen s o pe iod p, al, . . . , O P , o S, no in OM, he e exis s a dise ee d : X' -4 X' such ha he componen X i o M S inside á i is homeomo phic o X', and FPIXi - d . (e) (i) G- ( o TOP o)/q)/Po .  I q is e en, all bounda y componen s ha e pe iod q . I q is odd, all bounda y componen s ha e pe iod q o 2q . (ii) G- ( oR,/2)/I'o . All bounda y componen s o S ha e pe iod ei he 1 o 2 . BRAID TYPES WITH ZERO ENTROPY  54 9 In bo h pa s o case (c), o each o bi o bounda y componen s o pe iod p, 91 5 . . . , ap, o S no in OM, he e exis s a disc ee d : X' - X' such ha he componen Xi o M S inside Vi is homeomo phic o X' and F P I Xi - d i p is e en, o he e exis s a e e sing disc ee d' : X' - X' such ha he componen Xi o M S inside al is homeomo phic o X' and F P I Xi -d i p is odd . P oo .. We apply Theo em 1 o he comple ion G : T2 , T 2 o FAS . Case (a) G ^- TO P o)/q/I'o, so all poin s o T 2 ha e pe iod q unde G . Hence all bounda y componen s o S ha e pe iod q, and i we conside he o bi s a l , . . . . 0, o hose no in aM, he co esponding decomposi ion componen s Xi o M S inside al a e o genus ze o, and we may apply Theo em B, so he e exis s a disc ee d : X' -> X' such ha X i is homeomo phic o X', and F 4 1Xi - d . Case (b) G - R  ,/I'o, w= 7 /2, n : k = 4, 2, o G - Ru,/I'o, w= 7 /3, ±27 /3 : k = 6, 3 . The s a emen s abou he o bi s o bounda y componen s a e an immedia e consequence o applying Theo em 2 o he comple ion G : T 2 -> T 2 . As in case (a), since G is o- p, we ob ain disc ees in each o bi o bounda y componen s o S no in 8M . Case (c) (i) G- ( oTO P o)/ a )/I'o . F om he p oo o Theo em 1, he bounda y componen s o S ha e pe iod q i q is e en, o hey ha e pe iod q o 2q i q is odd . Case (c) (ü) G - ( o R, /2)/I'o . Again om he p oo o Theo em 1, he bounda y componen s o S ha e pe iod 1 o 2 . In bo h pa s o case (c), each componen o T 2 S inside a bounda y com- ponen no in 8M o e en pe iod con ains a disc ee since F is o - p, whils hose o odd pe iod ' con ain a e e sing disc ee since F ' is o- , by Theo em B . 55 0  J . GUASCHI, J . LLIBRE, R .S . MACKAY 4 . Two Co olla ies As a co olla y o he esul s o Sec ion 3 we ob ain he genus one case o [H1] : Theo em 5 . Le : M -M be an o- di eomo phism o a su ace M o genus one . I has pe iodic o bi s, o o bi s o bounda y componen s, wi h 3 dis inc odd pe os, hen h( ) > 0 . To de i e his om he abo e, we equi e he genus ze o esul o [BF], [H1], also de i ed by [LM1] : Theo em C . Le : X-X be an o - di eomo phism o a su ace o genus ze o . I has pe iodic o bi s o o bi s o bounda y componen s wi h wo dis inc odd pe os, hen h( ) > 0 . P oo o Theo em 5 : Le : M -~ M be an o- di eomo phism o a su ace M o genus one . Le P be he union o h ee o bi s o dis inc odd pe iod . Le F be a Thu s on canonical o m o P . Suppose h( ) -- 0 . Then he e a e wo cases : Case (a) : Suppose he e is a o a ional educing cu e, I', say, wi h pe iod p . Remo e i s annula neighbou hood and i s images, hen we ob ain a disjoin union o annuli A i . F om Theo em 3, he e a e wo possibili ies . In he i s case, he A i ha e pe iod p, so he Ai a e pe mu ed by F, and hus p di ides he o de o each pe iodic o bi , hence p is odd . So i we conside any Ai, F P ¡A¡ is o - , since is o - . Ai mus con ain bounda y componen s co esponding o he h ee o bi s o dis inc odd pe iod . So by Theo em C, h( ) > 0, a con adic ion . The o he possibili y is when p = 2, and he e a e wo in a ian annuli Ao, Al . Then one o Ao, Al mus con ain bounda y componen s co esponding o a leas wo o he h ee o bi s, and since FIA jj (i = 0,1) is o - , Theo em C implies ha h( ) > 0, a con adic ion . Case (b) : Suppose he e is no o a ional educing cu e . Then he e exis s a unique decomposi ion componen S o genus one . Le G : T2 -> T2 be he com- ple ion o FAS . Thenwe a e in Case (c) o Theo em 4 . I G- ( -T(P o)M/I'o, all bounda y componen s o S lla e pe iod q o 2q . I G- ( o R,/2)/I'o, hen all bounda y componen s o S lla e pe iod 1 o 2 . In pa icula , all bounda y componen s o S o odd pe iod lla e he same pe iod p, and he emaining bounda y componen s co esponding o P mus lie wi hin decomposi ion com- ponen s X i o genus ze o whose ou e bounda ies ha e pe iod p . Since F P I Xi is o- , and a leas one o he X i con ains bounda y componen s o odd o de , BRAID TYPES WITH ZERO ENTROPY  55 1 no p, co esponding o he o bi s o P, hen Theo em C implies ha h ( ) > 0, a con adic ion . As a second co olla y we will de i e a esul o [LM2], [H2] o di eomo - phisms o he o us iso opic o he iden i y . We e e he eade o [LM2] o de ini ions o li s, o a ion ec o s, e c . We ecall ha o a con inuous map : T2 , T2 wi h li : R2 , I 8 2 , i I is he g oup o in ege ansla ions y ,, : x 1--> x + m, xE R2, m E Z2, and i is homo opic o he iden i y, hen y = -y o all " y EF . Theo em 6 . Le : T 2 - 3 T 2 be a homeomo phism o he o us iso opic o he iden i y, an .d suppose h( ) = 0 . Then all o a ion ec o s associa ed wi h he pe iodic o bi s o a e collinea . P oo .. Suppose ha has a ini e union o pe iodic o bi s, wi h associa ed dis inc o a ion ec o s pi/gi, i = 1, . . . , N . Then om [LM2], since is homo opic o he iden i y, o each i E {1, . . . , N} he e exis s a pe iodic o bi Qá o p imi i e o a ion ype (pi, qi) . Le P =UN 1 Qá, and le F be a Thu s on canonical o m o P . Suppose F is o ini e o de , hen using (*) and Theo em 1, we see ha F - T(P o)1e/I'o, o some q E N, pE w, so all poin s o T 2 a e pe iodic wi h pe iod qand o a ion ec o (p, 0)/q . The emaining possibili y is ha F is educible (wi h ini e o de componen s, hough we shall no need his) . The e a e no non- o a ional educing cu es o F . Fo suppose F we e such a cu e, hen i mus su ound a leas wo holes, bu hey mus come om he same o bi , so he o a ion ype o ha o bi canno be p imi i e, which is a con adic ion . Suppose he e is a o a ional educing cu e 1' o F . Le G : T 2 -> T 2 be he comple ion o F, and le G : R2 --> R2 be a li o G . Le m E 71 2 {0} be he homo opy ype o 1', and q be i s pe iod . Then F li s o an in ini e se o cu es F, each in a ian unde T  ,,, which pa i ion he plane hi o in ini e s ips S each wi hin a bounded dis ance o some s aigh line o di ec ion m . Fu he mo e, he e exis s pE 71 2 such ha FqT F = 1', and since F is in e ible he same holds o he s ips S . Hence he o a ion se o F, and in pa icula he o a ion ec o s o he chosen pe iodic o bi s, a e con ained in he s aigh line {p/q + m : E IR} . This comple es he p oo . Appendix : P oo o Theo em 1 To p o e Theo em 1, we equi e he ollowing : 558  J . GUASCHI, J . LLIBRE, R .S . MACKAY [NS] V .V . NIKULIN AND I .R . SHAFAREVICH, "Geome ies and g oups," Sp inge , 1987 . [S] R .L.E . SCHWARZENBERGER, "N dimensional c ys allog aphy," Pi man esea ch no es in Ma hema ics 41, 1980 . [W] J.A . WOLF, "Spaces o cons an cu a u e," McG aw-Hill, 1967 . J . Guaschi : Nonlinea Sys ems Labo a o y Ma hema ics Ins i u e Uni e si y o Wa wick Co en y CV4 7AL ENGLAND J . Llib e : Depa amen de Ma emá iques Facul a de Ciéncies Uni e si a Au ónoma de Ba celona 08193 Bella e a (Ba celona) SPAIN R .S . Mackay : Nonlinea Sys ems Labo a o y Ma hema ics Ins i u e Uni e si y o Wa wick Co en y CV47AL ENGLAND P ime a e sió ebuda el 23 d'Agos de 1990, da e a e sió ebuda el 2 de Se emb e de 1991