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Pseudo solutions, rotation sets, and shadowing rotations for monotone recurrence relations

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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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References

  1. 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)

    Article  MathSciNet  Google Scholar 

  2. Angenent, S.: Monotone recurrence relations, their Birkhoff orbits and topological entropy. Ergod. Theory Dyn. Syst. 10, 15–41 (1990)

    Article  MathSciNet  Google Scholar 

  3. Aubry, S., Le Daeron, P.Y.: The discrete Frenkel–Kontorova model and its extensions. Physica D 8, 381–422 (1983)

    Article  MathSciNet  Google Scholar 

  4. 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)

    Chapter  Google Scholar 

  5. Barge, M., Swanson, R.: Rotation shadowing properties of circle and annulus maps. Ergod. Theory Dyn. Syst. 8, 509–521 (1988)

    Article  MathSciNet  Google Scholar 

  6. Blank, M.L.: Metric properties of minimal solutions of discrete periodical variational problems. Nonlinearity 2, 1–22 (1989)

    Article  MathSciNet  Google Scholar 

  7. 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)

    Chapter  Google Scholar 

  8. Boyland, P.L.: Rotation sets and Morse decompositions in twist maps. Ergod. Theory Dyn. Syst. 8, 33–61 (1988)

    Article  MathSciNet  Google Scholar 

  9. 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)

    Chapter  Google Scholar 

  10. Casdagli, M.: Periodic orbits for dissipative twist maps. Ergod. Theory Dyn. Syst. 7, 165–173 (1987)

    Article  MathSciNet  Google Scholar 

  11. Franks, J.: Recurrence and fixed points of surface homeomorphisms. Ergod. Theory Dyn. Syst. 8, 99–107 (1988)

    Article  MathSciNet  Google Scholar 

  12. 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)

    Article  MathSciNet  Google Scholar 

  13. Hall, G.R.: A topological version of a theorem of Mather on twist maps. Ergod. Theory Dyn. Syst. 4, 585–603 (1984)

    Article  MathSciNet  Google Scholar 

  14. Handel, M.: The rotation set of a homeomorphism of the annulus is closed. Commun. Math. Phys. 127, 339–349 (1990)

    Article  MathSciNet  Google Scholar 

  15. Ito, R.: Rotation sets are closed. Math. Proc. Camb. Philos. Soc. 89, 107–111 (1981)

    Article  MathSciNet  Google Scholar 

  16. Jenkinson, O.: Ergodic optimization. Discrete Contin. Dyn. Syst. 15, 197–224 (2006)

    Article  MathSciNet  Google Scholar 

  17. Jenkinson, O.: Ergodic optimization in dynamical systems. Ergod. Theory Dyn. Syst. 39, 2593–2618 (2019)

    Article  MathSciNet  Google Scholar 

  18. Katok, A.: Some remarks on the Birkhoff and Mather twist theorems. Ergod. Theory Dyn. Syst. 2, 183–194 (1982)

    MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MathSciNet  Google Scholar 

  20. Koropecki, A.: Realizing rotation numbers on annular continua. Math. Z. 285, 549–564 (2017)

    Article  MathSciNet  Google Scholar 

  21. Le Calvez, P.: Existence d’orbites quasi-periodiques dans les attracteurs de Birkhoff. Commun. Math. Phys. 106, 383–394 (1986)

    Article  Google Scholar 

  22. 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)

    Google Scholar 

  23. 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)

    Article  MathSciNet  Google Scholar 

  24. Miao, X.-Q., Qin, W.-X., Wang, Y.-N.: Secondary invariants of Birkhoff minimizers and heteroclinic orbits. J. Differ. Equ. 260, 1522–1557 (2016)

    Article  MathSciNet  Google Scholar 

  25. Misiurewicz, M.: Twist sets for maps of the circle. Ergod. Theory Dyn. Syst. 4, 391–404 (1984)

    Article  MathSciNet  Google Scholar 

  26. Mather, J.N.: Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology 21, 457–467 (1982)

    Article  MathSciNet  Google Scholar 

  27. Mather, J.N.: Variational construction of orbits of twist diffeomorphisms. J. Am. Math. Soc. 4, 207–263 (1991)

    Article  MathSciNet  Google Scholar 

  28. Mramor, B., Rink, B.: Ghost circles in lattice Aubry–Mather theory. J. Differ. Equ. 252, 3163–3208 (2012)

    Article  MathSciNet  Google Scholar 

  29. Newhouse, S., Palis, J., Takens, F.: Bifurcations and stability of families of diffeomorphisms. IHES Publ. Math. 57, 5–71 (1983)

    Article  MathSciNet  Google Scholar 

  30. Wang, K., Miao, X.-Q., Wang, Y.-N., Qin, W.-X.: Continuity of depinning force. Adv. Math. 335, 276–306 (2018)

    Article  MathSciNet  Google Scholar 

  31. Walters, P.: An Introduction to Ergodic Theory. Springer, New York (1982)

    Book  Google Scholar 

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Acknowledgements

We are grateful to the referee for some suggestions which help to improve the exposition of the paper.

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  1. Department of Mathematics, Soochow University, Suzhou, 215006, China

    Tong Zhou & Wen-Xin Qin

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  1. Tong Zhou
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  2. Wen-Xin Qin
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Correspondence to Wen-Xin Qin.

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Supported by the National Natural Science Foundation of China (11771316, 11790274).

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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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