Abstract
By introducing for monotone recurrence relations pseudo solutions, which are analogues of pseudo orbits of dynamical systems, we show that for general monotone recurrence relations the rotation set is closed, and each element in the rotation set is realized by a Birkhoff orbit. Moreover, if there is an orbit without rotation number, then the system has positive topological entropy, and we can construct orbits shadowing different rotation numbers.
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Alsedà, L., Llibre, J., Mañosas, F., Misiurewicz, M.: Lower bounds of the topological entropy for continuous maps of the circle of degree one. Nonlinearity 1, 463–479 (1988)
Angenent, S.: Monotone recurrence relations, their Birkhoff orbits and topological entropy. Ergod. Theory Dyn. Syst. 10, 15–41 (1990)
Aubry, S., Le Daeron, P.Y.: The discrete Frenkel–Kontorova model and its extensions. Physica D 8, 381–422 (1983)
Bangert, V.: Mather sets for twist maps and geodesics on tori. In: Kirchgraber, U., Walther, H.O. (eds.) Dynamics Reported, vol. 1, pp. 1–56. Wiley, New York (1988)
Barge, M., Swanson, R.: Rotation shadowing properties of circle and annulus maps. Ergod. Theory Dyn. Syst. 8, 509–521 (1988)
Blank, M.L.: Metric properties of minimal solutions of discrete periodical variational problems. Nonlinearity 2, 1–22 (1989)
Block, L., Guckenheimer, J., Misiurewicz, M., Young, L.-S.: Periodic points and topological entropy of one-dimensional maps. In: Nitecki, Z., Robinson, R.C. (eds.) Global Theory of Dynamical Systems, Springer Lecture Notes in Mathematics, vol. 819, pp. 18–34. Springer, New York (1980)
Boyland, P.L.: Rotation sets and Morse decompositions in twist maps. Ergod. Theory Dyn. Syst. 8, 33–61 (1988)
Boyland, P.L.: The rotation set as a dynamical invariant. In: McGehee, R., Meyer, K.R. (eds.) Twist Mappings and Their Applications, IMA Volumes in Mathematics, vol. 44, pp. 73–86. Springer, New York (1992)
Casdagli, M.: Periodic orbits for dissipative twist maps. Ergod. Theory Dyn. Syst. 7, 165–173 (1987)
Franks, J.: Recurrence and fixed points of surface homeomorphisms. Ergod. Theory Dyn. Syst. 8, 99–107 (1988)
Guo, L., Miao, X.-Q., Wang, Y.-N., Qin, W.-X.: Positive topological entropy for monotone recurrence relations. Ergod. Theory Dyn. Syst. 35, 1880–1901 (2015)
Hall, G.R.: A topological version of a theorem of Mather on twist maps. Ergod. Theory Dyn. Syst. 4, 585–603 (1984)
Handel, M.: The rotation set of a homeomorphism of the annulus is closed. Commun. Math. Phys. 127, 339–349 (1990)
Ito, R.: Rotation sets are closed. Math. Proc. Camb. Philos. Soc. 89, 107–111 (1981)
Jenkinson, O.: Ergodic optimization. Discrete Contin. Dyn. Syst. 15, 197–224 (2006)
Jenkinson, O.: Ergodic optimization in dynamical systems. Ergod. Theory Dyn. Syst. 39, 2593–2618 (2019)
Katok, A.: Some remarks on the Birkhoff and Mather twist theorems. Ergod. Theory Dyn. Syst. 2, 183–194 (1982)
Koch, H., de la Llave, R., Radin, C.: Aubry–Mather theory for functions on lattices. Discrete Contin. Dyn. Syst. (Ser. A) 3, 135–151 (1997)
Koropecki, A.: Realizing rotation numbers on annular continua. Math. Z. 285, 549–564 (2017)
Le Calvez, P.: Existence d’orbites quasi-periodiques dans les attracteurs de Birkhoff. Commun. Math. Phys. 106, 383–394 (1986)
Le Calvez, P.: Dynamical Properties of Diffeomorphisms of the Annulus and of the Torus, SMF/AMS Texts and Monographs, vol. 4. American Mathematical Society, Providence (2000)
Llibre, J., MacKay, R.S.: Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity. Ergod. Theory Dyn. Syst. 11, 115–128 (1991)
Miao, X.-Q., Qin, W.-X., Wang, Y.-N.: Secondary invariants of Birkhoff minimizers and heteroclinic orbits. J. Differ. Equ. 260, 1522–1557 (2016)
Misiurewicz, M.: Twist sets for maps of the circle. Ergod. Theory Dyn. Syst. 4, 391–404 (1984)
Mather, J.N.: Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology 21, 457–467 (1982)
Mather, J.N.: Variational construction of orbits of twist diffeomorphisms. J. Am. Math. Soc. 4, 207–263 (1991)
Mramor, B., Rink, B.: Ghost circles in lattice Aubry–Mather theory. J. Differ. Equ. 252, 3163–3208 (2012)
Newhouse, S., Palis, J., Takens, F.: Bifurcations and stability of families of diffeomorphisms. IHES Publ. Math. 57, 5–71 (1983)
Wang, K., Miao, X.-Q., Wang, Y.-N., Qin, W.-X.: Continuity of depinning force. Adv. Math. 335, 276–306 (2018)
Walters, P.: An Introduction to Ergodic Theory. Springer, New York (1982)
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Zhou, T., Qin, WX. Pseudo solutions, rotation sets, and shadowing rotations for monotone recurrence relations. Math. Z. 297, 1673–1692 (2021). https://doi.org/10.1007/s00209-020-02574-w
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DOI: https://doi.org/10.1007/s00209-020-02574-w