Revista Integración, temas de matemáticas.

Revista Integración, temas de matemáticas

Vol. 30 No. 1 (2012): Revista Integración, temas de matemáticas
Research and Innovation Articles

Collective dynamics

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  • Hector Mendez Lango
Hector Mendez Lango
UNAM, Facultad de Ciencias, Departamento de Matemáticas, Ciudad Universitaria, C.P. 04510, D.F., México

Published 2012-08-21

Keywords

  • Hyperspace,
  • discrete dynamics,
  • collective dynamics,
  • entropy

How to Cite

Mendez Lango, H. (2012). Collective dynamics. Revista Integración, Temas De matemáticas, 30(1), 25–41. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/2700

Abstract

For a metric compact setXand a continuous mapf: X→X we consider the hyperspace2X of all closed and nonempty subsets of X with the Hausdorff metric, and the induced mapˆf:2X→2X. In the past few years the study of the connection between the dynamical properties offand those Of ˆf has became an important and fruitful topic. In this paper we survey some significant results in this area. Also we collect some open questions and conjectures.

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References

  1. Acosta G., Illanes A. and Méndez-Lango H., “The transitivity of induced maps”, Topology Appl. 156 (2009), no. 5, 1013–1033.
  2. Adler R.L., Konheim A.G. and McAndrew M.H., “Topological entropy”, Trans. Amer. Math. Soc. 114 (1965), 309–319.
  3. Alseda L., Llibre J. and Misiurewicz M., Combinatorial dynamics and entropy in dimension one, Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co., River Edge, NJ, 1993.
  4. Arenas G., Isaacs R., Méndez H. y Sabogal S., Sistemas dinámicos discretos y fractales, Vínculos Matemáticos, 87, Departamento de Matemáticas, Facultad de Ciencias, UNAM, México, 2009.
  5. Banks J., “Topological mapping properties defined by digraphs”, Discrete Contin. Dynam. Systems 5 (1999), no. 1, 83–92.
  6. Banks J., “Chaos for induced hyperspace maps”, Chaos Solitons Fractals 25 (2005), no. 3, 1581–1583.
  7. Bauer W. and Sigmund K., “Topological dynamics of transformations induced on the space of probability measures”, Monatsh. Math. 79 (1975), 81–92.
  8. Block L.S. and Coppel W.A., “Dynamics in one dimension”, Lecture Notes in Mathematics, 1513, Springer-Verlag, Berlin, 1992.
  9. Devaney R.L., An introduction to chaotic dynamical systems, Second Edition, AddisonWesley Studies in Nonlinearity, Redwood City, CA, 1989.
  10. García J.L., Kwietniak D., Lampart M., Oprocha P. and Peris A., “Chaos on hyperspaces”, Nonlinear Anal. 71 (2009), no. 1-2, 1–8.
  11. Gengrong Z., Fanping Z. and Xinhe L., “Devaney’s chaotic on induced maps of hyperspace”, Chaos Solitons Fractals 27 (2006), no. 2, 471–475.
  12. Hernández P., Navegando en el hiperespacio, Tesis de Maestra en Ciencias (Matemáticas), UNAM, México, (2011).
  13. Hernández P., King J. and Méndez-Lango H., “Compact sets with dense orbit in 2X”, Topology Proc. 40 (2012), 319–330.
  14. Hocking J.G. and Young G.S., Topology, Dover Publications, New York, 1996.
  15. Illanes A., “Modelos de hiperespacios”, capítulo contenido en Invitación a la Teoría de los Continuos y sus Hiperespacios, Escobedo R., Macías S. y Méndez-Lango H., eds., Aportaciones Mat. Textos, 31, Soc. Mat. Mexicana, México, 2006.
  16. Illanes A. and Nadler S.B., Jr., Hyperspaces. Fundamentals and recent advances, Monographs and Textbooks in Pure and Applied Mathematics, 216. Marcel Dekker, Inc., New York, 1999.
  17. Kwietniak D. and Oprocha P., “Topological entropy and chaos for maps induced on hyperspaces”, Chaos Solitons Fractals 33 (2007), no. 1, 76–86.
  18. Lampart M. and Raith P., “Topological entropy for set-valued maps”, Nonlinear Anal. 73 (2010), no. 6, 1533–1537.
  19. Liao G., Wang L. and Zhang Y., “Transitivity, mixing and chaos for a class of set-valued mappings”, Sci. China Ser. A 49 (2006), no. 1, 1–8.
  20. Méndez-Lango H., “The process of finding f′ for an entire function f has infinite topological entropy”, Topology Proc. 28 (2004), no. 2, 639–646.
  21. Méndez-Lango H., “On density of periodic points for induced hyperspace maps”, Topology Proc. 35 (2010), 281–290.
  22. Nadler S.B., Jr. Dimension theory: An introduction with exercises, Aportaciones Mat. Textos, 18, Soc. Mat. Mexicana, México, 2002.
  23. Roman-Flores H., “A note on transitivity in set valued discrete systems”, Chaos Solitons Fractals 17 (2003), no. 1, 99–104.
  24. Schori R.M. and West J.E., “The hyperspace of the closed unit interval is a Hilbert Cube”, Trans. Amer. Math. Soc. 213 (1975), 217–235.
  25. Sharma P. and Nagar A., “Inducing sensitivity on hyperspaces”, Topology Appl. 157 (2010), no. 13, 2052–2058.
  26. Walters P., An introduction to ergodic theory, Grad. Texts in Math., 79, Springer Verlag, New York, 1982.
  27. Wang Y., Wei G. and Campbell W.H., “Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems”, Topology Appl. 156 (2009), no. 4, 803–811.