Let f be an orientation preserving homeomorphism of the disc D2 which
possesses a periodic point of period 3. Then either f is isotopic, relative the
periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2 pi
/3 or 4 pi /3, or f has a periodic point of least period n for each n in N*.Comment: 7 page

B Kolev

Blanchard P

Bowen R

Boyland P

Constantin A

Eilenberg S

Fathi A

Franks J

Gambaudo J M

Handel M

Jiang B

Katok A

Li T

Llibre J

Sarkovskii A

English

arXiv

7 pagesInternational audienceLet f be an orientation preserving homeomorphism of the disc D2 which possesses a periodic point of period 3. Then either f is isotopic, relative the periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2 pi /3 or 4 pi /3, or f has a periodic point of least period n for each n in N*

Kolev, Boris

HAL AMU

Periodic orbits of period 3 in the discBoris KolevTo cite this version:Boris Kolev. Periodic orbits of period 3 in the disc. Nonlinearity, IOP Publishing, 1994, 7 (3),pp.1067-1071. <10.1088/0951-7715/7/3/016>. <hal-00003272>HAL Id: hal-00003272https://hal.archives-ouvertes.fr/hal-00003272Submitted on 30 Nov 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.hal-00003272, version 1 - 30 Nov 2007PERIODIC ORBITS OF PERIOD 3 IN THE DISCBORIS KOLEVAbstract. Let f be an orientation preserving homeomorphismof the disc D2 which possesses a periodic point of period 3. Theneither f is isotopic, relative the periodic orbit, to a homeomorphismg which is conjugate to a rotation by 2pi/3 or 4pi/3, or J has aperiodic point of least period n for each n ∈ N∗.1. IntroductionSome very famous results of one dimensional dynamics state that acontinuous map of the interval which has a periodic point of period 3has periodic points of all other periods [16, 18] and positive topologicalentropy [3]. These results are certainly not true in higher dimensions;a rotation by 2π/3 in the disc D2 has periodic points of period 3 and1 only and topological entropy zero. In fact the period alone is notenough to specify a periodic orbit of a surface’s homeomorphism fin order to get information about the structure of the set of periodicpoints of f .What sort of stronger hypothesis do we have to add in order toget similar results in dimension two ? This problem has been exten-sively studied these last decades and a lot of results have emerged.Blanchard and Franks [1] have shown that an orientation reversinghomeomorphism f of the 2-sphere which has a point of period p anda point of period q where p and q are distinct odd integers has posi-tive topological entropy. If in addition, f is C1+ǫ [15], f has periodicpoints of arbitrary large period. Handel [12] has generalized this resultto orientation reversing homeomorphisms of any compact surface Musing Nielsen-Thurston theory. Following an idea of Bowen [2], theseauthors specify a periodic point z by the isotopy class of the restrictionfz of f to Mz = M − o(z, f); the complement in M of the orbit of zunder f . This idea has since been formalized by Boyland [4] who intro-duced the ‘braid type’ of a periodic orbit. Specifically, if [fz] denotesthe isotopy class of fz in Mz, define the braid type of o(z, f), denotedbt(z, f), to be the conjugacy class of [fz] in the mapping class group ofMz. Recently several authors [10, 17] have characterized all the braidDate: 28 January 1994.1991 Mathematics Subject Classification. 54H20, 58F13, 34C35.Key words and phrases. Dynamics of surfaces homeomorphisms; periodic orbits.12 BORIS KOLEVtypes which appear for diffeomorphisms of genus zero surfaces withtopological entropy zero.More recently, Gambaudo, van Strien and Tresser [11] have given anexample of a braid type of period 3 in the disc which imply all otherperiods and positive topological entropy, which is the exact analogueof the result in one dimensional dynamics stated previously. In thispaper we show that this is a quite general result, namely:Theorem 1.1. If f is an orientation preserving homeomorphism ofthe disc D2 which has a periodic point z of period 3, then either fis isotopic to a homeomorphism g which is conjugate to a rotation by2π/3 or 4π/3 or f has periodic points of least period n for each n ∈ N∗.In short terms, this result means that pseudo-Anosov 3 braid typesof the disc imply all other periods. By the way, this result is obviouslynot true for the only two other 3-braid types which are not pseudo-Anosov as cited previously. Therefore, it should be emphasized thatTheorem 1.1 is the best possible extension of the paper. “Period threeimplies chaos” [16] for the class of interval maps to the class of orien-tation preserving homeomorphisms of the disc D2.The proof of this theorem is based on the structure of pseudo-Anosovhomeomorphisms in the disc with three holes and upon some results ofJohn Franks about the existence of periodic points for homeomorphismsof the annulus [9]. In section 2, we recall the basic facts we need andgive the proof of Theorem 1.1 in section 3.2. Background and definitions2.1. Pseudo-Anosov homeomorphisms. We begin by recalling someproperties of pseudo-Anosov homeomorphisms that we need and thatcan be found in [7] or [8] (see also [14] for the definition of the fixed pointindex). A pseudo-Anosov homeomorphism φ of a compact orientablemanifold M with negative Euler characteristic is a homeomorphismsuch that there exists a pair of transverse measured foliations F u andF s and a number λ > 1 with φ(Fu) = λ−1Fu and φ(F s) = λF s.2.1.1. Each of the foliation have a finite number of singularities and atleast one on each component of the boundary ofM . These singularitiesare one-prong singularities on the boundary and p-prongs singularitieswith p ≥ 3 in the interior of M. Fu and F s have the same singulari-ties in the interior of M and on each component of the boundary thesingularities of Fu and F s alternate. Moreover, we have the followingformula for each foliation∑s(2− ps) = 2χ(M)PERIODIC ORBITS OF PERIOD 3 IN THE DISC 3where the sum is taken over all the singularities s, ps is the number ofprongs at s if s lies in the interior of M and ps = 3 if s is a one-prongsingularity on the boundary.2.1.2. Let x be a fixed point of φn. Then Ind(x, φn) 6= 0, Ind(x, φn) ≤1 and Ind(x, φn) < 0 if x lies on the boundary of M .2.1.3. Let x be a fixed point of φ.• If Ind(x, φ) = −1, then Ind(x, φn) = −1 for all n.• If Ind(x, φ) = 1, then there exist two positive integers p and qsuch that:Ind(x, φ) = Ind(x.φ2) = . . . = Ind(x, φq−1) = 1andInd(x, φq) = 1− p.In that case, there are exactly p stable (resp. unstable) half-leaves starting at x and the permutation induced by φ on theseshalf-leaves consists of p/q q-cycles (if p ≥ 3, x is a singularityof the foliations).2.1.4. If φ has a periodic point of least period n which is not on theboundary of M , then every map g isotopic to φ must possess a periodicpoint of least period n.2.1.5. φ has a dense orbit.2.2. Homeomorphisms of the annulus. Next, we briefly reviewsome results about homeomorphisms of the annulus and chain tran-sitivity [9]. Let f : A → A a homeomorphism of the annulus A =S1 × [0, 1] which is orientation preserving and boundary componentpreserving.Let π : A˜ → A be the universal cover, A˜ = R × [0, 1] andp1 : A˜ → R the projection onto the first coordinate.For each x ∈ A,choose a lift x˜ ∈ A˜ and consider:limn→∞p1(f˜n(x˜))− p1(x˜)nwhere f˜ is a lift of f . When this limit exists, it is called the f˜ -rotationnumber of x and denoted ρ(f˜ , x). It is independent of the choice ofthe lift x˜of x. If x belongs to one component of the boundary, ρ(x, f˜)is defined and independent of the choice of x on this component. Letρ1(f˜) and ρ2(f˜) these two numbers. A periodic point z is called a p/qperiodic point relative to f˜ if z has least period q and f˜ q(z˜) = z˜+(p, 0)for some lift (and hence all) z˜ of z.A homeomorphism f is called chain-transitive if for any x, y ∈ Aand ǫ > 0 there exists a sequence x0 = x, x1, . . . , xn = y, n ≥ 1 suchthat d(f(xi), xi+1) < ǫ. In particular if f has a dense orbit it is chaintransitive.4 BORIS KOLEVTheorem 2.1 (Franks). Let f : A→ A be an orientation preserving,boundary component preserving homeomorphism which is chain tran-sitive and f˜ : A˜→ A˜ a lift of f . Then for every rational p/q betweenρ1(f˜) and ρ2(f˜), f has a p/q periodic point.3. Proof of main theoremWe write zi = fi(z), (i = 1, 2, 3) for the points of the orbits of z andset o(z) = {z1, z2, z3}. Let Dz = D2 − o(z), M be the compact surfaceobtained from Dz by compactifying each puncture zi by a circle Si andπ : M → D2 be the map which collapses each circle Si to zi.There is a natural isomorphism between the mapping class group ofDz and the mapping class group of M . In our case [fz] is not reducible(in the sense of Nielsen-Thurston) because z is a point of period 3 andthere are no reducible braid type of prime period [4]. Consequently[fz] is either of finite order or of pseudo-Anosov type. If [fz] is of finiteorder, then there exists a homeomorphism g in D2 which is isotopicto f rel. o(z) and satisfy g3 = Id. Consequently, by a theorem ofKere´kja´rto´ [5, 6], g is conjugate to a rotation by 2π/3 or 4π/3 and weare done. If [fz] is of pseudo-Anosov type, we are going to show thatf possesses a periodic point of least period n for each integer n. Let φbe a pseudo-Anosov homeomorphism of M in the class [fz].Lemma 3.1. φ has a periodic point of least period n in the interior ofM for each n ∈ N∗\{3}.Proof of Lemma 3.1. χ(M) = −2 (M is a sphere with 4 holes !), so foreach foliation Fu and F s there is exactly one singularity on each bound-ary component of M and no singularity in the interior of M (2.1.1).Let S∞ be the component of ∂M corresponding to the boundary of D2.S∞ is invariant under φ and the singularities of the two foliations onS∞ are fixed points for φ (see Figure 1).Let L(φ) be the Lefschetz number of φ (see [14] for definition andcomputation of the Lefschetz number). We have:L(φ) =∑x∈F ix(φ)Ind(x, φ) = 1− Tr(φ∗1) = 1,where φ∗1 is the action induced by φ on the first homology group of M .Therefore, by (2.1.2), φ has a fixed point z0 such that Ind(z0, φ) = 1.It results from (2.1.2) that z0 lies in the interior of M and thus is not asingularity of the foliations. For the same reason (L(φ2) = 1), φ2 has aa fixed point of index 1 in the interior of M which is not a fixed pointof φ (2.1.3). Therefore φ has a periodic point of period 2 in the interiorof M .Blow up z0 into a circle S0 (there is no problem to extend the pseudo-Anosov φ to S0) and collapse the three circles S1, S2, S3 to the originalPERIODIC ORBITS OF PERIOD 3 IN THE DISC 5Figure 1. The manifold Mpoints z1, z2, z3. Using the fact that φ has a dense orbit (2.1.5), we ob-tain by this way a chain-transitive homeomorphism of a closed annulusA that we shall call Φ.Because z0 is not a singularity of the foliation, we have Ind(z0, φ2) =−1 by (2.1.3) and thus Φ has two period 2 periodic orbits on S0.Let Φ˜ be any lift of Φ to the universal cover A˜ of A and call ρ0(Φ˜)(resp. ρ∞(Φ˜)) the Φ˜-rotation number of Φ/s0 (resp. Φ/s∞). SinceΦ has a fixed point on S∞ and a periodic point of. period 2 on S0,ρ∞(Φ˜) = k for some k ∈ Z and ρ0(Φ˜) = (2l + 1)/2 for some l ∈ Z.Using Theorem 2.1 we conclude thatΦ has a periodic point of leastperiod n for each n ≥ 2. All the periodic points of period n ≥ 3belong to the interior of A and correspond therefore to periodic pointsof φ. Let us now give the proof of Theorem 1.1. Suppose first that f iscontinuously differentiable around the points of o(z). Then we canextend fz to a homeomorphism F : M →M which is isotopic to φ andthere is a one to one correspondence between periodic points of F inint(M) and periodic points of fZ in int(Dz). Using the previous lemmaand (2.1.4) f has a periodic point of least period n for every n ∈ N∗.In the case where f is not supposed to be well-behaved near o(z), weconstruct a sequence fk of homeomorphisms of D2 which coincide withf away from a l/k-neighbourhood of o(z), which act on {z1, z2, z3} asf but which are continuously differentiable around these points. Each6 BORIS KOLEVfk extends to a homeomorphism Fk of M which is isotopic to φ in Mand [13]:Lemma 3.2 (Handel). For every ǫ > 0, there exists δ > 0 (indepen-dent of k) such that if φ has a periodic point x of least period n andd(o(φ, x), ∂M) > ǫ then for every k, Fk has a periodic point xk of leastperiod n such that d(o(Fk, xk), ∂M) > δ.Therefore, f has a periodic point of every period. Remark. If we consider no longer a periodic point of period 3 buta periodic point of period q > 3 such that bt(z, f) is of type pseudo-Anosov and if the singularities of the foliations of a pseudo-Anosovrepresentative φ on S∞ are fixed points of φ, then the proof of Theo-rem 1.1 holds. There exists a fixed point z0 of index 1 for φ and suchthat when we blow up z0 into a circle S0 we get ρ(φ/S0) = k/l where kand l are coprime with 2 ≤ f ≤ q − 2. Therefore, we can prove thatf has periodic points of period n for every n ≥ q − 2. The generalsituation seems to be much more complex.References[1] P Blanchard and J Franks, The dynamical complexity of orientation-reversinghomeomorphisms of surfaces, Invent. Math. 62 (1980/81), no. 2, 333–339. MRMR595592 (82f:58073)[2] R Bowen, Entropy and the fundamental group, The structure of attractorsin dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D.,1977), Lecture Notes in Math., vol. 668, Springer, Berlin, 1978, pp. 21–29. MRMR518545 (80d:58049)[3] R Bowen and J Franks, The periodic points of maps of the disk and the interval,Topology 15 (1976), no. 4, 337–342. MR MR0431282 (55 #4283)[4] P Boyland, Topological methods in surface dynamics, Topology Appl. 58(1994), no. 3, 223–298. MR MR1288300 (95h:57016)[5] A Constantin and B Kolev, The theorem of Kere´kja´rto´ on periodic homeo-morphisms of the disc and the sphere, Enseign. Math. (2) 40 (1994), no. 3-4,193–204. MR MR1309126 (95j:55005)[6] S. Eilenberg, Sur les transformations periodiques de la surface de la sphe`re,Fund. Math. 22 (1934), 28–44.[7] A Fathi and F Laudenbach, The dynamics of the lift of a pseudo-Anoxov dif-feomorphism to the Poincare´ disc, in Papadopoulos Thesis, Orsay, 1983.[8] A Fathi, F Laudenbach, and V Poe´naru, Travaux de Thurston sur les surfaces,Asterisque, vol. 66-67, SMF, 1979.[9] J Franks, Recurrence and fixed points of surface homeomorphisms, ErgodicTheory Dynam. Systems 8∗ (1988), no. Charles Conley Memorial Issue, 99–107. MR MR967632 (90d:58124)[10] JM. Gambaudo, S van Strien, and C Tresser,The periodic orbit structure of ori-entation preserving diffeomorphisms on D2 with topological entropy zero, Ann.Inst. H. Poincare´ Phys. The´or. 50 (1989), no. 3, 335–356. MR MR1017968(90j:57008)[11] JM Gambaudo, S van Strien, and C Tresser, Vers un ordre de Sarkovski˘ı pourles plongements du disque pre´servant l’orientation, C. R. Acad. Sci. Paris Se´r.I Math. 310 (1990), no. 5, 291–294. MR MR1042866 (91f:58072)PERIODIC ORBITS OF PERIOD 3 IN THE DISC 7[12] M Handel, The entropy of orientation reversing homeomorphisms of surfaces,Topology 21 (1982), no. 3, 291–296. MR MR649760 (84f:58097)[13] , The rotation set of a homeomorphism of the annulus is closed, Comm.Math. Phys. 127 (1990), no. 2, 339–349. MR MR1037109 (91a:58102)[14] BJ Jiang, Lectures on Nielsen fixed point theory, Contemporary Mathemat-ics, vol. 14, American Mathematical Society, Providence, R.I., 1983. MRMR685755 (84f:55002)[15] A Katok, Lyapunov exponents, entropy and periodic orbits for diffeomor-phisms, Inst. Hautes E´tudes Sci. Publ. Math. (1980), no. 51, 137–173. MRMR573822 (81i:28022)[16] TY Li and JA. Yorke, Period three implies chaos, Amer. Math. Monthly 82(1975), no. 10, 985–992. MR MR0385028 (52 #5898)[17] J Llibre and RS MacKay, A classification of braid types for diffeomorphismsof surfaces of genus zero with topological entropy zero, J. London Math. Soc.(2) 42 (1990), no. 3, 562–576. MR MR1087229 (92a:57019)[18] OM Sarkovskii, Co-existence of cycles of a continuous mapping of the line intoitself, Ukrain. Mat. Z˘. 16 (1964), 61–71. MR MR0159905 (28 #3121)CMI, 39 rue F. Joliot-Curie, 13453 Marseille cedex 13, FranceE-mail address : kolev@cmi.univ-mrs.fr

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