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Continuum-Wise Expansive Homeomorphisms

Published online by Cambridge University Press:  20 November 2018

Hisao Kato
Hisao Kato*
Affiliation:
Faculty of Integrated Arts and Sciences, Hiroshima University, Hiroshima 730, Japan
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Abstract

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The notion of expansive homeomorphism is important in topological dynamics and continuum theory. In this paper, a new kind of homeomorphism will be introduced and studied, namely the continuum-wise expansive homeomorphism. The class of continuum-wise expansive homeomorphisms is much larger than the one of expansive homeomorphisms. In fact, the class of continuum-wise expansive homeomorphisms contains many important homeomorphisms which often appear in "chaotic" topological dynamics and continuum theory, but which are not expansive homeomorphisms. For example, the shift maps of Knaster's indecomposable chainable continua are continuum-wise expansive homeomorphisms, but they are not expansive homeomorphisms. Also, there is a continuum-wise expansive homeomorphism on the pseudoarc. We study several properties of continuum-wise expansive homeomorphisms. Many theorems concerning expansive homeomorphisms will be generalized to the case of continuum-wise expansive homeomorphisms.

Keywords

54F5054B2054H2054B2558F15expansive homeomorphisminverse limitshift mapsensitive dependence on inital conditionstopological entropytree-likecrookedindecomposable
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 3 , 01 June 1993 , pp. 576 - 598
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Aoki, N., Topological dynamics. In: Topics in general topology, (eds. K. Monta and J. Nagata), Elsevier Science Publishers B. V. (1989), 625–740.Google Scholar
2. Bing, R. H., Snake-like continua, Duke Math. J. 18(1951), 653–663.Google Scholar
3. Bryant, B. F., Unstable self-homeomorphisms of a compact space, Vanderbilt University, Thesis, 1954.Google Scholar
4. Fathi, A., Expansiveness, hyperbolicity and Hausdorff dimension, preprint.Google Scholar
5. Gottschalk, W., Minimal sets; an introduction to topological dynamics, Bull. Amer. Math. Soc. 64(1958), 336–351.Google Scholar
6. Gottschalk, W. and Hedlund, G., Topological dynamics, Amer. Math. Soc. Colloq. 34(1955).Google Scholar
7. Hiraide, K., Expansive homeomorphisms of compact surfaces are pseudo Anosov, Osaka J. Math. 27(1990), 117–162.Google Scholar
8. Hurewicz, and Wallman, , Dimension theory, Princeton Univ. Press, Princeton, N.J., 1948.Google Scholar
9. Jacobson, J. F. and Utz, W. R., The nonexistence of expansive homeomorphisms of a closed 2-cell, Pacific J. Math 10(1960), 1319–1321.Google Scholar
10. Kato, H., The nonexistence of expansive homeomorphisms of 1 -dimensional compact ANRs, Proc. Amer. Math. Soc. 108(1990), 267–269.Google Scholar
11. Kato, H., The nonexistence of expansive homeomorphisms ofPeano continua in the plane, Topology and its appl. 34(1990), 161–165.Google Scholar
12. Kato, H., The nonexistence of expansive homeomorphisms of Suslinian continua, J. Math. Soc. Japan, 42 (1990), 631–637.Google Scholar
13. Kato, H., On expansiveness of shift homeomorphisms of inverse limits of graphs, Fund. Math. 137(1991), 201–210.Google Scholar
14. Kato, H., The nonexistence of expansive homeomorphisms ofdendroids, Fund. Math. 136(1990), 37–43 .Google Scholar
15. Kato, H., Embeddability into the plane and movability on inverse limits of graphs whose shift maps are expansive, Topology and its appl. 43(1992), 141–156.Google Scholar
16. Kato, H., Expansive homeomorphisms in continuum theory, Topology and its appl., Proceedings of General Topology and Geometric Topology Symposium, (eds. Y. Kodama and T. Hoshina), 45(1992), 223–243.Google Scholar
17. Kato, H., Expansive homeomorphisms and indecomposability, Fund. Math. 139(1991), 49–57.Google Scholar
18. Kato, H. and Kawamura, K., A class of continua which admit no expansive homeomorphisms, Rocky Mountain J. Math, to appear.Google Scholar
19. Kennedy, J., A transitive homeomorphismon the pseudoarc which is semiconjugate to the tent map, Trans. Amer. Math. Soc, to appear.Google Scholar
20. Marié, R., Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc. 252(1979), 313–319.Google Scholar
21. Nadler, S. B., Jr., Hyperspaces of sets, Pure and Appl. Math. 49, Dekker, New York, 1978.Google Scholar
22. O'Brien, T. and Reddy, W., Each compact orientable surface of positive genus admits an expansive homeomorphism, Pacific J. Math 35(1970), 737–741.Google Scholar
23. Plykin, R. V., Sources and Sinks of A-Diffeomorphisms of Surfaces, Math. Sb. 23(1974), 233–253.Google Scholar
24. Plykin, R. V., On the geometry of hyperbolic attractors of smooth cascades, Russian Math. Survey 39(1984), 85–131.Google Scholar
25. Reddy, W., The existence of expansive homeomorphisms of manifolds, Duke Math. J. 32(1965), 627–632.Google Scholar
26. Reddy, W., Expansive canonical coordinates are hyperbolic, Topology and its appl. 15(1983), 205–210.Google Scholar
27. Walter, P., An introduction to ergodic theory, Graduate Texts in Math. 79, Springer.Google Scholar
28. Williams, R. F., A note on unstable homeomorphisms, Proc. Amer. Math. Soc. 6(1955), 308–309.Google Scholar