Glasnik Matematicki, Vol. 51, No. 2 (2016), 453-474.

DYNAMIC PROPERTIES FOR THE INDUCED MAPS ON n-FOLD SYMMETRIC PRODUCT SUSPENSIONS

Franco Barragán, Alicia Santiago-Santos and Jesús F. Tenorio

Instituto de Física y Matemáticas, Universidad Tecnológica de la Mixteca, Carretera a Acatlima, Km. 2.5, Huajuapan de León, Oaxaca, C.P. 69000, México
e-mail: franco@mixteco.utm.mx
e-mail: alicia@mixteco.utm.mx
e-mail: jtenorio@mixteco.utm.mx

Abstract.   Let X be a continuum. For any positive integer n we consider the hyperspace Fn(X) and if n is greater than or equal to two, we consider the quotient space SFn(X) defined in [3]. For a given map f:X → X, we consider the induced maps Fn(f): Fn(X) → Fn(X) and SFn(f): SFn(X) → SFn(X) defined in [4]. Let M be one of the following classes of maps: exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive, chaotic, minimal, irreducible, feebly open and turbulent. In this paper we study the relationships between the following statements: f M, Fn(f) M and SFn(f) M.

2010 Mathematics Subject Classification.   54B20, 37B45, 54F50, 54F15.

Key words and phrases.   Chaotic map, exact map, feebly open map, hyperspace, induced map, irreducible map, minimal map, mixing map, strongly transitive map, symmetric product, symmetric product suspension, totally transitive map, transitive map, turbulent map, weakly mixing map.

Full text (PDF) (free access)

DOI: 10.3336/gm.51.2.12

References:

  1. G. Acosta, A. Illanes and H. Méndez-Lango, The transitivity of induced maps, Topology Appl. 156 (2009), 1013-1033.
    MathSciNet     CrossRef

  2. J. Banks, Chaos for induced hyperspace maps, Chaos Solitons Fractals 25 (2005), 681-685.
    MathSciNet     CrossRef

  3. F. Barragán, On the n-fold symmetric product suspensions of a continuum, Topology Appl. 157 (2010), 597-604.
    MathSciNet     CrossRef

  4. F. Barragán, Induced maps on n-fold symmetric product suspensions, Topology Appl. 158 (2011), 1192-1205.
    MathSciNet     CrossRef

  5. F. Barragán, Aposyndetic properties of the n-fold symmetric product suspensions of a continuum, Glas. Mat. Ser. III 49(69) (2014), 179-193.
    MathSciNet     CrossRef

  6. F. Barragán, S. Macías and J. F. Tenorio, More on induced maps on n-fold symmetric product suspensions, Glas. Mat. Ser. III 50(70) (2015), 489-512.
    MathSciNet     CrossRef

  7. J. Camargo, C. García and A. Ramírez, Transitivity of the induced map Cn(f), Rev. Colombiana Mat. 48 (2014), 235-245.
    MathSciNet     CrossRef

  8. J. S. Cánovas-Peña and G. S. López, Topological entropy for induced hyperspace maps, Chaos Solitons Fractals 28 (2006), 979-982.
    MathSciNet     CrossRef

  9. J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1966.
    MathSciNet    

  10. J. L. Gómez-Rueda, A. Illanes and H. Méndez-Lango, Dynamic properties for the induced maps in the symmetric products, Chaos Solitons Fractals 45 (2012), 1180-1187.
    MathSciNet     CrossRef

  11. G. Higuera and A. Illanes, Induced mappings on symmetric products, Topology Proc. 37 (2011), 367-401.
    MathSciNet    

  12. R. Gu and W. Guo, On mixing property in set-valued discrete systems, Chaos Solitons Fractals 28 (2006), 747-754.
    MathSciNet     CrossRef

  13. A. Illanes and S. B. Nadler, Jr., Hyperspaces. Fundamentals and recent advances, Marcel Dekker, New York, 1999.
    MathSciNet    

  14. S. Kolyada, L. Snoha and S. Trofimchuk, Noninvertible minimal maps, Fund. Math. 168 (2001), 141-163.
    MathSciNet     CrossRef

  15. D. Kwietniak and M. Misiurewicz, Exact Devaney chaos and entropy, Qual. Theory Dyn. Syst. 6 (2005), 169-179.
    MathSciNet     CrossRef

  16. X. Ma, B. Hou and G. Liao, Chaos in hyperspace system, Chaos Solitons Fractals 40 (2009), 653-660.
    MathSciNet     CrossRef

  17. S. Macías, Topics on continua, Chapman & Hall/CRC, Boca Raton, 2005.
    MathSciNet     CrossRef

  18. S. B. Nadler, Jr., Hyperspaces of sets, Marcel Dekker, New York-Basel, 1978. Reprinted by: Sociedad Matemática Mexicana, México, 2006.
    MathSciNet    
    MathSciNet    

  19. S. B. Nadler, Jr., Continuum theory. An introduction, Marcel Dekker, New York, 1992.
    MathSciNet    

  20. A. Peris, Set-valued discrete chaos, Chaos Solitons Fractals 26 (2005), 19-23.
    MathSciNet     CrossRef

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

  22. M. Sabbaghan and H. Damerchiloo, A note on periodic points and transitive maps, Math. Sci. Q. J. 5 (2011), 259-266.
    MathSciNet    

  23. Y. Wang and G. Wei, Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems, Topology Appl. 155 (2007), 56-68.
    MathSciNet     CrossRef
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