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The depth of centres of maps on dendrites

Part of: Connections with other structures, applications

Published online by Cambridge University Press:  09 April 2009

Hisao Kato
Hisao Kato
Affiliation:
Institute of Mathematics University of Tsukuba Ibaraki 305 Japan e-mail: [email protected]
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Abstract

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Xiong proved that if f: I → I is any map of the unit interval I, then the depth of the centre of f is at most 2, and Ye proved that for any map f: T → T of a finite tree T, the depth of the centre of f is at most 3. It is natural to ask whether the result can be dendrites. In this note, we show that there is dendrite D such that for any countable ordinal number λ there is a map f: D →D such that the depth of centre of f is λ. As a corollary, we show that for any countable ordinal number λ there is a map (respectively a homeomorphism) f of a 2-dimensional ball B2 (respectively a 3-dimensional ball B3) such that the depth of centre of f is λ.

Keywords

Dendritenon-wandering pointscentredepth.

MSC classification

Secondary: 54H20: Topological dynamics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 1 , February 1998 , pp. 44 - 53
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Kato, H., ‘A note on periodic points and recurrent points of maps of dendrites’, Bull. Austral. Math. Soc. 51 (1995), 459–461.CrossRefGoogle Scholar
[2]Krasinkiewicz, J., On a method of constructing ANR-sets, An application of inverse limits’, Fund. Math. 92 (1976), 95–112.CrossRefGoogle Scholar
[3]Moise, E. E., Geometric topology in dimension 2 and 3, Graduate Texts in Math. 47 (Springer, Berlin, 1977).CrossRefGoogle Scholar
[4]Nadler, S. B. Jr, Continuum theory, Pure Appl. Math. 158 (Wiley, New York, 1992).Google Scholar
[5]Neumann, D. A., ‘Central sequences in dynamical systems’, Amer J. Math. 100 (1978), 1–18.CrossRefGoogle Scholar
[6]Xiong, J. C., ‘ for every continuous self-map f of the interval’, Kexue Tongbao 28 (1983), 21–23.Google Scholar
[7]Ye, X. D., ‘The center and the depth of the center of a tree map’, Bull. Austrol. Math. Soc. 48 (1993), 347–350.CrossRefGoogle Scholar