Rotation intervals and entropy on attracting annular continua

Passeggi, Alejandro¹ ; Potrie, Rafael¹ ; Sambarino, Martín¹

¹ Facultad de Ciencias, Centro de Matemática, Universidad de la República, Montevideo, Uruguay

Geometry & topology, Tome 22 (2018) no. 4, pp. 2145-2186.

Résumé

We show that if $f$ is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of $f$ is positive. Further, the entropy is shown to be associated to a $C^0$–robust rotational horseshoe. On the other hand, we construct examples of annular homeomorphisms with such attractors for which the rotation interval is uniformly large but the entropy approaches zero as much as desired.

The developed techniques allow us to obtain similar results in the context of Birkhoff attractors.

Détail

DOI : 10.2140/gt.2018.22.2145

Classification : 37E30, 37B40, 37B45, 37E45, 54H20
Keywords: rotation number, entropy, annular continua, surface homeomorphisms, horseshoes

Bibliographie

[1] R H Abraham, H B Stewart, A chaotic blue sky catastrophe in forced relaxation oscillations, Phys. D 21 (1986) 394 | DOI

[2] M U Akhmet, M O Fen, Entrainment by chaos, J. Nonlinear Sci. 24 (2014) 411 | DOI

[3] L Alsedà, J Llibre, F Mañosas, M Misiurewicz, Lower bounds of the topological entropy for continuous maps of the circle of degree one, Nonlinearity 1 (1988) 463 | DOI

[4] M C Arnaud, C Bonatti, S Crovisier, Dynamiques symplectiques génériques, Ergodic Theory Dynam. Systems 25 (2005) 1401 | DOI

[5] M Barge, R M Gillette, Rotation and periodicity in plane separating continua, Ergodic Theory Dynam. Systems 11 (1991) 619 | DOI

[6] F Beguín, Ensembles de rotations des homéomorphismes du tore T2, lecture notes (2007)

[7] G D Birkhoff, Sur quelques courbes fermées remarquables, Bull. Soc. Math. France 60 (1932) 1

[8] C Bonatti, S Crovisier, Récurrence et généricité, Invent. Math. 158 (2004) 33 | DOI

[9] C Bonatti, L J Díaz, E R Pujals, A C1–generic dichotomy for diffeomorphisms : weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. 158 (2003) 355 | DOI

[10] J P Boroński, P Oprocha, Rotational chaos and strange attractors on the 2–torus, Math. Z. 279 (2015) 689 | DOI

[11] H W Broer, KAM theory : the legacy of A N Kolmogorov’s 1954 paper, Bull. Amer. Math. Soc. 41 (2004) 507 | DOI

[12] A Chenciner, Poincaré and the three-body problem, from: "Henri Poincaré, 1912–2012" (editors B Duplantier, V Rivasseau), Prog. Math. Phys. 67, Springer (2015) 51 | DOI

[13] S Crovisier, Perturbation of C1–diffeomorphisms and generic conservative dynamics on surfaces, from: "Dynamique des difféomorphismes conservatifs des surfaces : un point de vue topologique", Panor. Synthèses 21, Soc. Math. France (2006) 1

[14] J Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc. 158 (1971) 301 | DOI

[15] J Franks, Rotation vectors for surface diffeomorphisms, from: "Proceedings of the International Congress of Mathematicians" (editor S D Chatterji), Birkhäuser (1995) 1179

[16] J Franks, M Handel, Entropy zero area preserving diffeomorphisms of S2, Geom. Topol. 16 (2012) 2187 | DOI

[17] J Franks, P Le Calvez, Regions of instability for non-twist maps, Ergodic Theory Dynam. Systems 23 (2003) 111 | DOI

[18] M Girard, Sur les courbes invariantes par un difféomorphisme C1–générique symplectique d’une surface, PhD thesis, Université d’Avignon (2009)

[19] J G Hocking, G S Young, Topology, Addison-Wesley (1961)

[20] T Jäger, Linearization of conservative toral homeomorphisms, Invent. Math. 176 (2009) 601 | DOI

[21] T Jäger, A Passeggi, On torus homeomorphisms semiconjugate to irrational rotations, Ergodic Theory Dynam. Systems 35 (2015) 2114 | DOI

[22] A Katok, B Hasselblatt, Introduction to the modern theory of dynamical systems, 54, Cambridge Univ. Press (1995) | DOI

[23] J Kennedy, J A Yorke, Topological horseshoes, Trans. Amer. Math. Soc. 353 (2001) 2513 | DOI

[24] A Koropecki, Realizing rotation numbers on annular continua, Math. Z. 285 (2017) 549 | DOI

[25] A Koropecki, P Le Calvez, M Nassiri, Prime ends rotation numbers and periodic points, Duke Math. J. 164 (2015) 403 | DOI

[26] A Koropecki, A Passeggi, A Poincaré–Bendixson theorem for translation lines and applications to prime ends, preprint (2017)

[27] J M Kwapisz, Rotation sets and entropy, PhD thesis, State University of New York at Stony Brook (1995)

[28] P Le Calvez, Propriétés des attracteurs de Birkhoff, Ergodic Theory Dynam. Systems 8 (1988) 241 | DOI

[29] P Le Calvez, Une version feuilletée équivariante du théorème de translation de Brouwer, Publ. Math. Inst. Hautes Études Sci. 102 (2005) 1 | DOI

[30] P Le Calvez, F A Tal, Forcing theory for transverse trajectories of surface homeomorphisms, preprint (2015)

[31] J Llibre, R S Mackay, Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity, Ergodic Theory Dynam. Systems 11 (1991) 115 | DOI

[32] J N Mather, Topological proofs of some purely topological consequences of Carathéodory’s theory of prime ends, from: "Selected studies: physics-astrophysics, mathematics, history of science" (editors T M Rassias, G M Rassias), North-Holland (1982) 225

[33] S Matsumoto, Prime end rotation numbers of invariant separating continua of annular homeomorphisms, Proc. Amer. Math. Soc. 140 (2012) 839 | DOI

[34] J W Milnor, Attractor, Scholarpedia 1 (2006) | DOI

[35] M Misiurewicz, K Ziemian, Rotation sets for maps of tori, J. London Math. Soc. 40 (1989) 490 | DOI

[36] M H A Newman, Elements of the topology of plane sets of points, Cambridge Univ. Press (1951)

[37] A Passeggi, Contributions in surface dynamics: a classification of minimal sets of homeomorphisms and aspects of the rotation theory on the torus, PhD thesis, Friedrich-Alexander-Universität, Erlangen (2013)

[38] A Passeggi, J Xavier, A classification of minimal sets for surface homeomorphisms, Math. Z. 278 (2014) 1153 | DOI

[39] R Shaw, Strange attractors, chaotic behavior, and information flow, Z. Naturforsch. A 36 (1981) 80 | DOI

[40] M Shub, All, most, some differentiable dynamical systems, from: "Proceedings of the International Congress of Mathematicians" (editors M Sanz-Solé, J Soria, J L Varona, J Verdera), Eur. Math. Soc. (2006) 99

[41] J M T Thompson, H B Stewart, Nonlinear dynamics and chaos, Wiley (2002)

[42] R B Walker, Periodicity and decomposability of basin boundaries with irrational maps on prime ends, Trans. Amer. Math. Soc. 324 (1991) 303 | DOI