Dynamics of covering maps of the annulus I: semiconjugacies. (English) Zbl 1367.37041
Summary: It is often the case that a covering map of the open annulus is semiconjugate to a map of the circle of the same degree. We investigate this possibility and its consequences on the dynamics. In particular, we address the problem of the classification up to conjugacy. However, there are examples which are not semiconjugate to a map of the circle, and this opens new questions.
MSC:
| 37E10 | Dynamical systems involving maps of the circle |
| 37C15 | Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems |
References:
| [1] | Anosov, D.V., Katok, A.B.: New examples in smooth ergodic theory. Ergodic diffeomorphisms. Trans. Mosc. Math. Soc. 23, 1-35 (1970) · Zbl 0255.58007 |
| [2] | Béguin, F., Crovisier, S., LeRoux, F., Patou, A.: Pseudo-rotations of the closed annulus: variation on a theorem of J. Kwapisz. Nonlinearity 17, 1427-1453 (2004) · Zbl 1077.37032 · doi:10.1088/0951-7715/17/4/016 |
| [3] | Béguin, F., Crovisier, S., LeRoux, F.: Pseudo-rotations of the open annulus. Bull. Braz. Math. Soc. 37(2), 275-306 (2006) · Zbl 1105.37029 · doi:10.1007/s00574-006-0013-2 |
| [4] | Bamón, R., Kiwi, J., Rivera-Letelier, J., Urzúa, R.: On the topology of solenoidal attractors of the cylinder. Ann. I. H. Poincaré 23, 209-236 (2006) · Zbl 1092.37017 · doi:10.1016/j.anihpc.2005.03.002 |
| [5] | de Melo, W., van Strien, S.: One-Dimensional Dynamics. Springer, Berlin (1993) · Zbl 0791.58003 · doi:10.1007/978-3-642-78043-1 |
| [6] | Fathi, A., Herman, M.: Existence de difféomorphismes minimaux. Asterisque 49, 37-59 (1977) · Zbl 0374.58010 |
| [7] | Fayad, B., Katok, A.: Constructions in elliptic dynamics. Ergod. Theory Dyn. Syst. 24(5), 1477-1520 (2004) · Zbl 1089.37012 · doi:10.1017/S0143385703000798 |
| [8] | Fayad, B., Saprykina, M.: Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary. Ann. Sci. École Norm. Sup. (4) 38, 339-364 (2005) · Zbl 1090.37001 |
| [9] | Handel, M.: A pathological area preserving \[C^\infty C\]∞ diffeomorphism of the plane. Proc. Am. Math. Soc. 86(1), 163-168 (1982) · Zbl 0509.58031 |
| [10] | Herman, Michael: Construction of some curious diffeomorphisms of the Riemann sphere. J. Lond. Math. Soc. (2) 34(2), 375-384 (1986) · Zbl 0603.58017 · doi:10.1112/jlms/s2-34.2.375 |
| [11] | Iglesias, J., Portela, A., Rovella, A.: Structurally stable perturbations of polynomials in the Riemann sphere. Ann. de l’Inst. H. Poincaré 25(6), 1209-1220 (2009) · Zbl 1153.37329 · doi:10.1016/j.anihpc.2008.02.001 |
| [12] | Iglesias, J., Portela, A., Rovella, A., Xavier, J.: Attracting sets on surfaces. Proc. Am. Math. Soc. 143(2):765-779 (2015) · Zbl 1346.37024 |
| [13] | Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle. J. Math. Pure. Appl. 4, 167-244 (1885) · JFM 17.0680.01 |
| [14] | Przytycki, F.: On \[\Omega\] Ω-stability and structural stability of endomorphisms satisfying Axiom A. Studia Math. 60, 61-77 (1977) · Zbl 0343.58008 |
| [15] | Tsujii, M.: Fat solenoidal attractors. Nonlinearity 14, 1011-1027 (2001) · Zbl 1067.37028 · doi:10.1088/0951-7715/14/5/306 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
