Dynamic properties of the dynamical system SFnm(X), SFnm(f))

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


Let X be a continuum and let n be a positive integer. We consider the hyperspaces Fn(X) and SFn(X). If m is an integer such that n > m ≥ 1, we consider the quotient space SFnm(X). For a given map f : X → X, we consider the induced maps Fn(f) : Fn(X) → Fn(X), SFn(f) : SFn(X) → SFn(X) and SFnm(f) : SFnm(X) → SFnm(X). In this paper, we introduce the dynamical system (SFnm(X), SFnm (f)) and we investigate some relationships between the dynamical systems (X, f), (Fn(X), Fn(f)), (SFn(X), SFn(f)) and (SFnm(X), SFnm(f)) when these systems are: exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive, chaotic, irreducible, feebly open and turbulent.


chaotic; continuum; dynamical system; exact; feebly open; hyperspace; induced map; irreducible; mixing; strongly transitive; symmetric product; symmetric product suspension; totally transitive; transitive; turbulent; weakly mixing

Subject classification

54B20; 37B45; 54F50; 54F15.

Full Text:



G. Acosta, A. Illanes and H. Méndez-Lango, The transitivity of induced maps, Topology Appl. 156, no. 5 (2009), 1013-1033. https://doi.org/10.1016/j.topol.2008.12.025

E. Akin, The General Topology of Dynamical Systems, Grad. Stud. Math., vol. 1, Amer. Math. Soc., Providence, 1993.

J. Banks, Chaos for induced hyperspace maps, Chaos Solitons Fractals 25, no. 3 (2005), 681-685. https://doi.org/10.1016/j.chaos.2004.11.089

F. Barragán, On the n-fold symmetric product suspensions of a continuum, Topology Appl. 157, no. 3 (2010), 597-604. https://doi.org/10.1016/j.topol.2009.10.017

F. Barragán, Induced maps on n-fold symmetric product suspensions, Topology Appl. 158, no. 10 (2011), 1192-1205. https://doi.org/10.1016/j.topol.2011.04.006

F. Barragán, Aposyndetic properties of the n-fold symmetric product suspensions of a continuum, Glas. Mat. Ser. III 49(69), no. 1 (2014), 179-193. https://doi.org/10.3336/gm.49.1.13

F. Barragán, S. Macías and J. F. Tenorio, More on induced maps on n-fold symmetric product suspensions, Glas. Mat. Ser. III 50(70), no. 2 (2015), 489-512. https://doi.org/10.3336/gm.50.2.15

F. Barragán, A. Santiago-Santos and J. F. Tenorio, Dynamic properties for the induced maps on n-fold symmetric product suspensions, Glas. Mat. Ser. III 51(71), no. 2 (2016), 453-474.

W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math. 79 (1975), 81-92. https://doi.org/10.1007/BF01585664

G. D. Birkhoff, Dynamical Systems, American Math. Soc., Colloquium Publication., Vol. IX, Amer. Math. Soc. Providence, R. I., 1927.

K. Borsuk and S. Ulam, On symmetric products of topological space, Bull. Amer. Math. Soc. 37, no. 12 (1931), 875-882. https://doi.org/10.1090/S0002-9904-1931-05290-3

J. Camargo, C. García and A. Ramírez, Transitivity of the Induced Map $C_n(f)$, Rev. Colombiana Mat. 48, no. 2 (2014), 235-245. https://doi.org/10.15446/recolma.v48n2.54131

J. S. Cánovas-Peña and G. Soler-López, Topological entropy for induced hyperspace maps, Chaos Solitons Fractals 28, no. 4 (2006), 979-982. https://doi.org/10.1016/j.chaos.2005.08.173

E. Castañeda-Alvarado and J. Sánchez-Martínez, On the unicoherence of $F_n(X)$ and $SF^n_m(X)$ of continua, Topology Proc. 42 (2013), 309-326.

E. Castañeda-Alvarado, F. Orozco-Zitli and J. Sánchez-Martínez, Induced mappings between quotient spaces of symmetric products of continua, Topology Appl. 163 (2014), 66-76. https://doi.org/10.1016/j.topol.2013.10.007

J. Dugundji, Topology, Boston, London, Sydney, Toronto: Allyn and Bacon, Inc, 1966.

L. Fernández and C. Good, Shadowing for induced maps of hyperspaces, Fund. Math. 235, no. 3 (2016), 277-286. https://doi.org/10.4064/fm136-2-2016

J. L. Gómez-Rueda, A. Illanes and H. Méndez-Lango, Dynamic properties for the induced maps in the symmetric products, Chaos Solitons Fractals 45, no. 9-10 (2012), 1180-1187. https://doi.org/10.1016/j.chaos.2012.05.003

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

A. Illanes and S. B. Nadler, Jr., Hyperspaces: fundamentals and recent advances, Monographs and Textbooks in Pure and Applied Math., Vol. 216, Marcel Dekker, New York, Basel, 1999.

S. Kolyada, L. Snoha and S. Trofimchuk, Noninvertible minimal maps, Fund. Math. 168, no. 2 (2001), 141-163. https://doi.org/10.4064/fm168-2-5

D. Kwietniak, Exact Devaney chaos and entropy, Qual. Theory Dyn. Syst. 6, no. 1 (2005), 169-179. https://doi.org/10.1007/BF02972670

D. Kwietniak and P. Oprocha, Topological entropy and chaos for maps induced on hyperspaces, Chaos Solitons Fractals 33, no. 1 (2007), 76-86. https://doi.org/10.1016/j.chaos.2005.12.033

G. Liao, L. Wang and Y. Zhang, Transitivity, mixing and chaos for a class of set-valued mappings, Sci. China Ser. A 49, no. 1 (2006), 1-8. https://doi.org/10.1007/s11425-004-5234-5

X. Ma, B. Hou and G. Liao, Chaos in hyperspace system, Chaos Solitons Fractals 40, no. 2 (2009), 653-660. https://doi.org/10.1016/j.chaos.2007.08.009

J. C. Macías, On n-fold pseudo-hyperspace suspensions of continua, Glas. Mat. Ser. III 43(63), no. 2 (2008), 439-449. https://doi.org/10.3336/gm.43.2.14

S. Macías, On the n-fold hyperspaces suspension of continua, Topology Appl. 138, no. 1-3 (2004), 125-138. https://doi.org/10.1016/j.topol.2003.08.023

S. Macías, Topics on Continua, Pure and Applied Mathematics Series, Vol. 275, Chapman & Hall/CRC, Taylor & Francis Group, Boca Raton, London, New York, Singapore, 2005.

S. B. Nadler, Jr., A fixed point theorem for hyperspaces suspensions, Houston J. Math. 5, no. 1 (1979), 125-132.

S. B. Nadler, Jr., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, New York, Basel, 1978. Reprinted in: Aportaciones Matemáticas de la Sociedad Matemática Mexicana, Serie Textos #33, 2006.

A. Peris, Set-valued discrete chaos, Chaos Solitons Fractals 26, no. 1 (2005), 19-23. https://doi.org/10.1016/j.chaos.2004.12.039

H. Román-Flores, A note on transitivity in set-valued discrete systems, Chaos Solitons Fractals 17, no. 1 (2003), 99-104. https://doi.org/10.1016/S0960-0779(02)00406-X

M. Sabbaghan and H. Damerchiloo, A note on periodic points and transitive maps, Math. Sci. Q. J. 5, no. 3 (2011), 259-266.

Y. Wang and G. Wei, Characterizing mixing, weak mixing and transitivity on induced hyperspace dynamical systems, Topology Appl. 155, no. 1 (2007), 56-68. https://doi.org/10.1016/j.topol.2007.09.003

Abstract Views

Metrics Loading ...

Metrics powered by PLOS ALM

Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt