Glasnik Matematicki, Vol. 59, No. 2 (2024), 479-505. \( \)

MINIMAL DYNAMICAL SYSTEMS WITH CLOSED RELATIONS

Iztok Banič, Goran Erceg, Rene Gril Rogina and Judy Kennedy

Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, SI-2000 Maribor, Slovenia, Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia, Andrej Marušič Institute, University of Primorska, Muzejski trg 2, SI-6000 Koper, Slovenia
e-mail:iztok.banic@um.si

Faculty of Science, University of Split, Rudera Boškovića 33, Split, Croatia
e-mail:goran.erceg@pmfst.hr

Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, SI-2000 Maribor, Slovenia
e-mail:rene.gril@student.um.si

Lamar University, 200 Lucas Building, P.O. Box 10047, Beaumont, TX 77710, USA
e-mail:kennedy9905@gmail.com

Abstract.   We introduce dynamical systems \((X,G)\) with closed relations \(G\) on compact metric spaces \(X\) and discuss different types of minimality of such dynamical systems, all of them generalizing minimal dynamical systems \((X,f)\) with continuous function \(f\) on a compact metric space \(X\).

2020 Mathematics Subject Classification.   54C60, 54F15, 54F17

Key words and phrases.   Closed relations, dynamical systems, minimal dynamical systems, \(CR\)-dynamical systems, minimal \(CR\)-dynamical systems, backward minimal \(CR\)-dynamical systems, invariant sets, forward orbits, backward orbits, omega limit sets, alpha limit sets, topological conjugations

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https://doi.org/10.3336/gm.59.2.10

References:

  1. E. Akin, General topology of dynamical systems, American Mathematical Society, Providence, 1993.
    MathSciNet    CrossRef

  2. I. Banič, G. Erceg and J. Kennedy, Closed relations with non-zero entropy that generate no periodic points, Discrete Contin. Dyn. Syst. 42 (2022), 5137–5166.
    MathSciNet    CrossRef

  3. I. Banič, G. Erceg, S. Greenwood and J. Kennedy, Transitive points in CR-dynamical systems, Topology Appl. 326 (2023), No. 108407.
    MathSciNet    CrossRef

  4. G. D. Birkhoff, Quelques theoremes sur le mouvement des systemes dynamiques, Bull. Soc. Math. France 40 (1912), 305–323.
    MathSciNet    Link

  5. L. Christiano and S. Harrison, Chaos, sunspots and automatic stabilizers, J. Monetary Economics 44 (1999), 3–31.

  6. W. Cordeiro and M. J. Pacfico, Continuum-wise expansiveness and specification for set-valued functions and topological entropy, Proc. Amer. Math. Soc. 144 (2016), 4261–4271.
    MathSciNet    CrossRef

  7. K. Kawamura and J. Kennedy, Shift maps and their variants on inverse limits with set-valued functions, Topology Appl. 239 (2018), 92–114.
    MathSciNet    CrossRef

  8. S. Kolyada and L. Snoha, Minimal dynamical systems, Scholarpedia 4 (2009), 5803.
    CrossRef

  9. S. Kolyada, L. Snoha and S. Tromchuk, Noninvertible minimal maps, Fund. Math. 168 (2001), 141–163.
    MathSciNet    CrossRef

  10. W. T. Ingram, An introduction to inverse limits with set-valued functions, Springer, New York, 2012.
    MathSciNet    CrossRef

  11. W. T. Ingram and W. S. Mahavier, Inverse limits of upper semi-continuous set valued functions, Houston J. Math. 32 (2006), 119–130.
    MathSciNet

  12. I. D. Woods and J. P. Kelly, Chaotic dynamics in family of set-valued functions, Minnesota Journal of Undergraduate Mathematics 3 (2018), 1–19.
    Link

  13. J. A. Kennedy, and V. Nall, Dynamical properties of shift maps on Inverse limits with a set valued map, Ergodic Theory Dynam. Systems 38 (2018), 1499–1524.
    MathSciNet    CrossRef

  14. A. Loranty and R. J. Pawlak, On the transitivity of multifunctions and density of orbits in generalized topological spaces, Acta Math. Hungar. 135 (2012), 56–66.
    MathSciNet    CrossRef

  15. J. Li, K. Yan, and X. Ye, Recurrence properties and disjointness on the induced spaces, Discrete Contin. Dyn. Syst. 35 (2015), 1059–1073.
    MathSciNet    CrossRef

  16. G. Liao, L. Wang, and Y. Zhang, Transitivity, mixing and chaos for a class of set-valued mappings, Sci. China Ser. A 49 (2006), 1–8.
    MathSciNet    CrossRef

  17. P. Maličky, Backward orbits of transitive maps, J. Difference Equ. Appl. 18 (2012), 1193–1203.
    MathSciNet    CrossRef

  18. W. S. Mahavier, Inverse limits with subsets of \([0,1]\times [0,1]\), Topology Appl. 141 (2004), 225–231.
    MathSciNet    CrossRef

  19. R. Metzger, C. A. Morales Rojas and P. Thieullen, Topological stability in set-valued dynamics, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), 1965–1975.
    MathSciNet    CrossRef

  20. H. Román-Flores, A note on transitivity in set-valued discrete systems, Chaos Solitons Fractals 17 (2003), 99–104.
    MathSciNet    CrossRef

  21. K. Sang Wong, and Z. Salleh, Topologically transitive and mixing properties of set-valued dynamical systems, Abstr. Appl. Anal. (2021), no. 5541105.
    MathSciNet    CrossRef
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