Abstract and Applied Analysis

Asymptotic Properties of G -Expansive Homeomorphisms on a Metric G -Space

Ruchi Das and Tarun Das

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Abstract

We define and study the notions of positively and negatively G -asymptotic points for a homeomorphism on a metric G -space. We obtain necessary and sufficient conditions for two points to be positively/negatively G -asymptotic. Also, we show that the problem of studying G -expansive homeomorphisms on a bounded subset of a normed linear G -space is equivalent to the problem of studying linear G -expansive homeomorphisms on a bounded subset of another normed linear G -space.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 237820, 10 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475909

Digital Object Identifier
doi:10.1155/2012/237820

Mathematical Reviews number (MathSciNet)
MR2994931

Zentralblatt MATH identifier
1262.54012

Citation

Das, Ruchi; Das, Tarun. Asymptotic Properties of $G$ -Expansive Homeomorphisms on a Metric $G$ -Space. Abstr. Appl. Anal. 2012 (2012), Article ID 237820, 10 pages. doi:10.1155/2012/237820. https://projecteuclid.org/euclid.aaa/1364475909


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References

  • W. R. Utz, “Unstable homeomorphisms,” Proceedings of the American Mathematical Society, vol. 1, pp. 769–774, 1950.
  • B. F. Bryant, “On expansive homeomorphisms,” Pacific Journal of Mathematics, vol. 10, pp. 1163–1167, 1960.
  • B. F. Bryant, “Expansive Self-Homeomorphisms of a Compact Metric Space,” The American Mathematical Monthly, vol. 69, no. 5, pp. 386–391, 1962.
  • J. D. Wine, “Nonwandering sets, periodicity, and expansive homeomorphisms,” Topology Proceedings, vol. 13, no. 2, pp. 385–395, 1988.
  • R. K. Williams, “Further results on expansive mappings,” Proceedings of the American Mathematical Society, vol. 58, pp. 284–288, 1976.
  • R. K. Williams, “Linearization of expansive homeomorphisms,” General Topology and its Applications, vol. 6, no. 3, pp. 315–318, 1976.
  • N. G. Markley, “Homeomorphisms of the circle without periodic points,” Proceedings of the London Mathematical Society. Third Series, vol. 20, pp. 688–698, 1970.
  • B. F. Bryant and P. Walters, “Asymptotic properties of expansive homeomorphisms,” Mathematical Systems Theory, vol. 3, pp. 60–66, 1969.
  • M. Eisenberg and J. H. Hedlund, “Expansive automorphisms of Banach spaces,” Pacific Journal of Mathematics, vol. 34, pp. 647–656, 1970.
  • J. H. Hedlund, “Expansive automorphisms of Banach spaces. II,” Pacific Journal of Mathematics, vol. 36, pp. 671–675, 1971.
  • R. Das, “Expansive self-homeomorphisms on \emphG-spaces,” Periodica Mathematica Hungarica, vol. 31, no. 2, pp. 123–130, 1995.
  • T. Choi and J. Kim, “Decomposition theorem on \emphG-spaces,” Osaka Journal of Mathematics, vol. 46, no. 1, pp. 87–104, 2009.
  • N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, vol. 52 of North-Holland Mathematical Library, North-Holland, Amsterdam, The Netherlands, 1994.
  • R. Das, “On \emphG-expansive homeomorphisms and generators,” The Journal of the Indian Mathematical Society. New Series, vol. 72, no. 1–4, pp. 83–89, 2005.
  • R. Das and T. K. Das, “On extension of \emphG-expansive homeomorphisms,” The Journal of the Indian Mathematical Society. New Series, vol. 67, no. 1–4, pp. 35–41, 2000.
  • R. Das and T. Das, “On properties of G-expansive homeomorphisms,” Mathematica Slovaca, vol. 62, no. 3, pp. 531–538, 2012.
  • R. Das and T. Das, “Topological transitivity of uniform limit functions on \emphG-spaces,” International Journal of Mathematical Analysis, vol. 6, no. 29–32, pp. 1491–1499, 2012.
  • E. Shah and T. K. Das, “On pseudo orbit tracing property in \emphG-spaces,” JP Journal of Geometry and Topology, vol. 3, no. 2, pp. 101–112, 2003.
  • G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York, NY, USA, 1972.
  • R. S. Palais, “The classification of \emphG-spaces,” Memoirs of the American Mathematical SocietY, vol. 36, pp. 1–71, 1960.
  • R. Das and T. Das, “A note on representation of pseudovariant maps,” Mathematica Slovaca, vol. 62, no. 1, pp. 137–142, 2012.
  • H. B. Keynes and J. B. Robertson, “Generators for topological entropy and expansiveness,” Mathematical Systems Theory, vol. 3, pp. 51–59, 1969.