# Hyperspaces and the S-equivariant Complete Invariance Property

Maury, Saurabh Chandra

• Accepted : 2014.04.21
• Published : 2015.03.23
• 18 9

#### Abstract

In this paper it is investigated as to when a nonempty invariant closed subset A of a $S^1$-space X containing the set of stationary points (S) can be the fixed point set of an equivariant continuous selfmap on X and such space X is said to possess the S-equivariant complete invariance property (S-ECIP). It is also shown that if X is a metric space and $S^1$ acts on $X{\times}S^1$ by the action $(x,p){\cdot}q=(x,p{\cdot}q)$, where p, $q{\in}S^1$ and $x{\in}X$, then the hyperspace $2^{X{\times}S^1}$ of all nonempty compact subsets of $X{\times}S^1$ has the S-ECIP.

#### Keywords

Equivariant map;Hyperspaces;Hausdorff metric;CIP;CIPH

#### References

1. S. Antonyan, West's problem on equivariant hyperspaces and Banach-Mazur compacta, Trans. Amer. Math. Soc., 355(2003), 3379-3404. https://doi.org/10.1090/S0002-9947-03-03217-3
2. K. K. Azad and K. Srivastava, On S-equivariant complete invariance property, Journal of the Indian Math. Soc., 62(1996), 2005-2009.
3. T. Banakh, R. Voytsitskyy, Characterizing metric spaces whose hyperspaces are absolute neighborhood retracts, Topology Appl., 154(2007), 2009-2025. https://doi.org/10.1016/j.topol.2006.02.009
4. G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York, London, 1972.
5. A. Chigogidze, K. H. Hofmann and J. R. Martin, Compact groups and fixed point sets, Trans. Amer. Math. Soc., 349(1997), 4537-4554. https://doi.org/10.1090/S0002-9947-97-02059-X
6. D. W. Curtis, Hyperspaces of noncompact metric spaces, Compositio Mathematica, 40(1980), 126-130.
7. J. R. Martin, Fixed point sets of homeomorphisms of metric products, Proc. Amer. Math. Soc., 103(1988), 1293-1298. https://doi.org/10.1090/S0002-9939-1988-0955025-9
8. J. R. Martin and S. B. Nadler Jr., Examples and questions in the theory of fixed point sets, Canad. J. Math., 31(1997), 1017-1032.
9. J. R. Martin, L. G. Oversteegen and E. D. Tymchatyn, Fixed point sets of products and cones, Pacific J. Math., 101(1982), 133-139. https://doi.org/10.2140/pjm.1982.101.133
10. J. R. Martin and W. A. R. Weiss, Fixed point sets of metric and nonmetric spaces, Trans. Amer. Math. Soc., 284(1984), 337-353. https://doi.org/10.1090/S0002-9947-1984-0742428-1
11. S. B. Nadler Jr., Hyperspaces of sets: A text with research questions, Monographs and Textbooks, Pure Appl. Math. 49, Marcel Dekker, New York and Basel, 1978.
12. H. Schirmer, Fixed point sets of continuous selfmaps, in: Fixed Point Theory, Conf. Proc., Sherbrooke, 1980, Lecture Notes in Math., 866(1981), 417-428.
13. H. Schirmer, Fixed point sets of homeomorphisms of compact surfaces, Israel J. Math., 10(1971), 373-378. https://doi.org/10.1007/BF02771655
14. H. Schirmer, On fixed point sets of homeomorphisms of the n-ball, Israel J. Math., 7(1969), 46-50. https://doi.org/10.1007/BF02771745
15. L. E. Ward Jr., Fixed point sets, Pacific J. Math., 47(1973), 553-565. https://doi.org/10.2140/pjm.1973.47.553

#### Acknowledgement

Supported by : CSIR