Topological horseshoes for surface homeomorphisms. (English) Zbl 1518.37054
The authors study surface homeomorphisms isotopic to the identity. The main result of the paper is a purely topological criterion to ensure the existence of topological horseshoes. They present several applications of this criterion for homeomorphisms of genus zero surfaces with zero topological entropy. For instance, they show that every point of the open annulus with non-empty \(\omega\)-limit has a rotation number and this rotation number is unique if the whole annulus is a generalized region of instability. In addition, a complete study of sphere homeomorphisms without horseshoes is undertaken.
Reviewer: Héctor Barge (Madrid)
MSC:
| 37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |
| 37E35 | Flows on surfaces |
| 37E45 | Rotation numbers and vectors |
| 37B40 | Topological entropy |
References:
| [1] | S. B. ANGENENT, “A remark on the topological entropy and invariant circles of an area preserving twistmap” in Twist Mappings and Their Applications, IMA Vol. Math. Appl. 44, Springer, New York, 1992, 1-5. · Zbl 0769.58036 · doi:10.1007/978-1-4613-9257-6_1 |
| [2] | M. C. ARNAUD, Création de connexions en topologie \[{C^1} \], Ergodic Theory Dynam. Systems 21 (2001), no. 2, 339-381. · Zbl 0997.37007 · doi:10.1017/S0143385701001183 |
| [3] | M. BARGE and R. M. GILLETTE, Rotation and periodicity in plane separating continua, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 619-631. · Zbl 0756.54021 · doi:10.1017/S0143385700006398 |
| [4] | F. BÉGUIN, S. CROVISIER, and F. LE ROUX, Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: The Denjoy-Rees technique, Ann. Sci. Éc. Norm. Supér. (4) 40 (2007), no. 2, 251-308. · Zbl 1132.37003 · doi:10.1016/j.ansens.2007.01.001 |
| [5] | F. BÉGUIN, S. CROVISIER, and F. LE ROUX, Fixed point sets of isotopies on surfaces, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 6, 1971-2046. · Zbl 1442.37058 · doi:10.4171/jems/960 |
| [6] | A. S. BESICOVITCH, A problem on topological transformations of the plane, II, Proc. Cambridge Philos. Soc. 47 (1951), 38-45. · Zbl 0054.07004 · doi:10.1017/s0305004100026347 |
| [7] | G. D. BIRKHOFF, Sur quelques courbes fermées remarquables, Bull. Soc. Math. France 60 (1932), 1-26. · JFM 58.0633.01 · doi:10.24033/bsmf.1182 |
| [8] | C. BONATTI, J.-M. GAMBAUDO, J.-M. LION, and C. TRESSER, Wandering domains for infinitely renormalizable diffeomorphisms of the disk, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1273-1278. · Zbl 0843.57016 · doi:10.2307/2161198 |
| [9] | P. BOYLAND, Rotation sets and monotone periodic orbits for annulus homeomorphisms, Comment. Math. Helv. 67 (1992), no. 2, 203-213. · Zbl 0763.58012 · doi:10.1007/BF02566496 |
| [10] | P. BOYLAND and G. R. HALL, Invariant circles and the order structure of periodic orbits in monotone twist maps, Topology 26 (1987), no. 1, 21-35. · Zbl 0618.58032 · doi:10.1016/0040-9383(87)90017-6 |
| [11] | J. CONEJEROS, The local rotation set is an interval, Ergodic Theory Dynam. Systems 38 (2018), no. 7, 2571-2617. · Zbl 1402.37055 · doi:10.1017/etds.2016.129 |
| [12] | S. CROVISIER, Periodic orbits and chain-transitive sets of \[{\mathcal{C}^1} -diffeomorphisms \], Publ. Math. Inst. Hautes Études Sci. 104 (2006), 87-141. · Zbl 1226.37007 · doi:10.1007/s10240-006-0002-4 |
| [13] | S. CROVISIER, A. KOCSARD, A. KOROPECKI, and E. PUJALS, Structure of attractors for strongly dissipative surface diffeomorphisms, in preparation. |
| [14] | S. CROVISIER, E. PUJALS, and C. TRESSER, Boundary of chaos and period doubling for dissipative embeddings of the 2-dimensional disk, in preparation. |
| [15] | A. DE CARVALHO, M. LYUBICH, and M. MARTENS, Renormalization in the Hénon family, I: Universality but non-rigidity, J. Stat. Phys. 121 (2005), no. 5-6, 611-669. · Zbl 1098.37039 · doi:10.1007/s10955-005-8668-4 |
| [16] | J. FRANKS, Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math. (2) 128 (1988), no. 1, 139-151. · Zbl 0676.58037 · doi:10.2307/1971464 |
| [17] | J. FRANKS, “Rotation numbers for \[{\mathbb{S}^2}\] diffeomorphisms” in Modern Theory of Dynamical Systems, Contemp. Math., 692, Amer. Math. Soc., Providence, 2017, 101-110. · Zbl 1385.37059 · doi:10.1090/conm/692/13913 |
| [18] | J. FRANKS and M. HANDEL, Entropy zero area preserving diffeomorphisms of \[{\mathbb{S}^2} \], Geom. Topol. 16 (2012), no. 4, 2187-2284. · Zbl 1359.37039 · doi:10.2140/gt.2012.16.2187 |
| [19] | M. HANDEL, Zero entropy surface diffeomorphisms (1986), preprint. |
| [20] | P. HAZARD, M. MARTENS, and C. TRESSER, Infinitely many moduli of stability at the dissipative boundary of chaos, Trans. Amer. Math. Soc. 370 (2018), no. 1, 27-51. · Zbl 1416.37024 · doi:10.1090/tran/6940 |
| [21] | T. HOMMA, An extension of the Jordan curve theorem, Yokohama Math. J. 1 (1953), 125-129. · Zbl 0051.40104 |
| [22] | O. JAULENT, Existence d’un feuilletage positivement transverse à un homéomorphisme de surface, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 4, 1441-1476. · Zbl 1352.37126 |
| [23] | A. KATOK, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. 51 (1980), no. 4, 137-173. · Zbl 0445.58015 |
| [24] | J. KENNEDY and J. A. YORKE, Topological horseshoes, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2513-2530. · Zbl 0972.37011 · doi:10.1090/S0002-9947-01-02586-7 |
| [25] | A. KOROPECKI, Realizing rotation numbers on annular continua, Math. Z. 285 (2007), no. 1-2, 549-564. · Zbl 1361.37047 · doi:10.1007/s00209-016-1720-z |
| [26] | P. LE CALVEZ, Une version feuilletée équivariante du théorème de translation de Brouwer, Publ. Math. Inst. Hautes Études Sci. 102 (2005), 1-98. · Zbl 1152.37015 · doi:10.1007/s10240-005-0034-1 |
| [27] | P. LE CALVEZ, Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 1, 141-176. · Zbl 1168.37010 · doi:10.24033/asens.2065 |
| [28] | P. LE CALVEZ and F. A. TAL, Forcing theory for transverse trajectories of surface homeomorphisms, Invent. Math. 212 (2018), no. 2, 619-729. · Zbl 1390.37073 · doi:10.1007/s00222-017-0773-x |
| [29] | P. LE CALVEZ and J.-C. YOCCOZ, Un théorème d’indice pour les homéomorphismes du plan au voisinage d’un point fixe, Ann. of Math. (2) 146 (1997), no. 2, 241-293. · Zbl 0895.58032 · doi:10.2307/2952463 |
| [30] | P. LE CALVEZ and J.-C. YOCCOZ, Suite des indices de Lefschetz des itérées pour un domaine de Jordan qui est un bloc isolant, preprint, 1997, http://matheuscmss.wordpress.com/2019/07/04/suite-des-indices-de-lefschetz-des-iteres-pour-un-domaine-de-jordan-qui-est-un-bloc-isolant/ . |
| [31] | F. LE ROUX, L’ensemble de rotation autour d’un point fixe, Astérisque 350, Soc. Math. France, Paris, 2013. · Zbl 1286.37002 |
| [32] | M. LYUBICH and M. MARTENS, Renormalization in the Hénon family, II: The heteroclinic web, Invent. Math. 186 (2011), no. 1, 115-189. · Zbl 1243.37035 · doi:10.1007/s00222-011-0316-9 |
| [33] | A. PASSEGGI, R. POTRIE, and M. SAMBARINO, Rotation intervals and entropy on attracting annular continua, Geom. Topol. 22 (2018), no. 4, 2145-2186. · Zbl 1386.37042 · doi:10.2140/gt.2018.22.2145 |
| [34] | R. PÉREZ-MARCO, Fixed points and circle maps, Acta Math. 179 (1997), no. 2, 243-294. · Zbl 0914.58027 · doi:10.1007/BF02392745 |
| [35] | M. REES, A minimal positive entropy homeomorphism of the 2-torus, J. Lond. Math. Soc. (2) 23 (1981), no. 3, 537-550. · Zbl 0451.58022 · doi:10.1112/jlms/s2-23.3.537 |
| [36] | S. SMALE, Differentiable dynamical systems, Bull. Amer. Math. Soc. (N.S.) 73 (1967), 747-817. · Zbl 0202.55202 · doi:10.1090/S0002-9904-1967-11798-1 |
| [37] | A. SZYMCZAK, The Conley index and symbolic dynamics, Topology 35 (1996), no. 2, 287-299. · Zbl 0855.58023 · doi:10.1016/0040-9383(95)00029-1 |
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