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Positive topological entropy for monotone recurrence relations

Published online by Cambridge University Press:  30 June 2014

LI GUO ,
XUE-QING MIAO ,
YA-NAN WANG  and
WEN-XIN QIN
LI GUO
Affiliation:
Department of Mathematics, Soochow University, Suzhou, 215006, China email qinwx@suda.edu.cn
XUE-QING MIAO
Affiliation:
Department of Mathematics, Soochow University, Suzhou, 215006, China email qinwx@suda.edu.cn
YA-NAN WANG
Affiliation:
Department of Mathematics, Soochow University, Suzhou, 215006, China email qinwx@suda.edu.cn
WEN-XIN QIN
Affiliation:
Department of Mathematics, Soochow University, Suzhou, 215006, China email qinwx@suda.edu.cn
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Abstract

We associate the topological entropy of monotone recurrence relations with the Aubry–Mather theory. If there exists an interval $[{\it\rho}_{0},{\it\rho}_{1}]$ such that, for each ${\it\omega}\in ({\it\rho}_{0},{\it\rho}_{1})$, all Birkhoff minimizers with rotation number ${\it\omega}$ do not form a foliation, then the diffeomorphism on the high-dimensional cylinder defined via the monotone recurrence relation has positive topological entropy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 6 , September 2015 , pp. 1880 - 1901
Copyright
© Cambridge University Press, 2014 

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References

Angenent, S.. Monotone recurrence relations, their Birkhoff orbits and topological entropy. Ergod. Th. & Dynam. Sys.(10) (1990), 15–41.CrossRefGoogle Scholar
Angenent, S.. A remark on the topological entropy and invariant circles of an area preserving twist map. Twist Mappings and Their Applications (IMA Volumes in Mathematics, 44). Eds. McGehee, R. and Meyer, K. R.. Springer, New York, 1992, pp. 1–5.Google Scholar
Aubry, S. and Abramovici, G.. Chaotic trajectories in the standard map, the concept of anti-integrability. Physica D 43 (1990), 199–219.CrossRefGoogle Scholar
Aubry, S. and Le Daeron, P. Y.. The discrete Frenkel–Kontorova model and its extensions. Phys. D 8 (1983), 381–422.CrossRefGoogle Scholar
Bangert, V.. A uniqueness theorem for ℤn -periodic variational problems. Comment. Math. Helv. 62 (1987), 511–531.CrossRefGoogle Scholar
Bangert, V.. Mather sets for twist maps and geodesics on tori. Dynamics Reported. Vol. 1. Eds. Kirchgraber, U. and Walther, H. O.. Wiley, New York, 1988, pp. 1–56.Google Scholar
Bangert, V.. On minimal laminations of the torus. Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), 95–138.CrossRefGoogle Scholar
Burns, K. and Weiss, H.. A geometric criterion for positive topological entropy. Commun. Math. Phys. 172 (1995), 95–118.CrossRefGoogle Scholar
Blank, M. L.. Metric properties of minimal solutions of discrete periodical variational problems. Nonlinearity 2 (1989), 1–22.CrossRefGoogle Scholar
Boyland, P. and Hall, G. R.. Invariant circles and the order structure of periodic orbits in monotone twist maps. Topology 26 (1987), 21–35.CrossRefGoogle Scholar
Chen, Y.-C.. On topological entropy of billiard tables with small inner scatterers. Adv. Math. 224 (2010), 432–460.CrossRefGoogle Scholar
de la Llave, R. and Valdinoci, E.. Ground states and critical points for generalized Frenkel–Kontorova models in ℤd. Nonlinearity 20 (2007), 2409–2424.CrossRefGoogle Scholar
de la Llave, R. and Valdinoci, E.. Ground states and critical points for Aubry–Mather theory in statistical mechanics. J. Nonlinear Sci. 20 (2010), 153–218.CrossRefGoogle Scholar
Knauf, A., Schulz, F. and Siburg, K. F.. Positive topological entropy for multi-bump magnetic fields. Nonlinearity 26 (2013), 727–743.CrossRefGoogle Scholar
Koch, H., de la Llave, R. and Radin, C. Aubry–Mather theory for functions on lattices. Discrete Contin. Dyn. Syst. (Ser. A) 3 (1997), 135–151.CrossRefGoogle Scholar
Li, M.-C. and Lyu, M.-J.. Positive topological entropy for multidimensional perturbations of topologically crossing homoclinicity. Discrete Contin. Dyn. Syst. (Ser. A) 30 (2011), 243–252.CrossRefGoogle Scholar
Mather, J.. Variational construction of orbits of twist diffeomorphisms. J. Amer. Math. Soc. 4 (1991), 207–263.CrossRefGoogle Scholar
Moser, J.. Minimal solutions of variational problems on a torus. Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), 229–272.CrossRefGoogle Scholar
Mramor, B. and Rink, B.. Ghost circles in lattice Aubry–Mather theory. J. Differential Equations 252 (2012), 3163–3208.CrossRefGoogle Scholar
Mramor, B. and Rink, B.. A dichotomy theorem for minimizers of monotone recurrence relations. Ergod. Th. & Dynam. Sys. doi: http://dx.doi.org/10.1017/etds.2013.47, published online 27 September 2013.CrossRefGoogle Scholar
Mramor, B.. Monotone variational recurrence relations. PhD Thesis, VU University, Amsterdam, 2012.Google Scholar
Slijepčević, S.. The pulled Frenkel–Kontorova chain. Nonlinearity 11 (1998), 923–948.CrossRefGoogle Scholar
Slijepčević, S.. Monotone gradient dynamics and Mather’s shadowing. Nonlinearity 12 (1999), 969–986.CrossRefGoogle Scholar
Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 1982.CrossRefGoogle Scholar