Extending expansive homeomorphisms
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- by J. D. Wine
- Proc. Amer. Math. Soc. 86 (1982), 531-534
- DOI: https://doi.org/10.1090/S0002-9939-1982-0671230-8
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Abstract:
In previous work on extensions of expansive homeomorphisms there has always been a finiteness condition placed on the remainder. In this note we give a necessary and sufficient condition for a self-homeomorphism to be expansive by restrictions on orbits, and then use this to give a sufficient condition for the homeomorphic extension of an expansive homeomorphism to be expansive. Two examples are given wherein the homeomorphic extensions of expansive homeomorphism are examined relative to the conditions of the theorem.References
- B. F. Bryant, On expansive homeomorphisms, Pacific J. Math. 10 (1960), 1163–1167. MR 120632
- B. F. Bryant, Expansive Self-Homeomorphisms of a Compact Metric Space, Amer. Math. Monthly 69 (1962), no. 5, 386–391. MR 1531676, DOI 10.2307/2312129
- Walter Helbig Gottschalk and Gustav Arnold Hedlund, Topological dynamics, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R.I., 1955. MR 0074810
- J. F. Jakobsen and W. R. Utz, The non-existence of expansive homeomorphisms on a closed $2$-cell, Pacific J. Math. 10 (1960), 1319–1321. MR 117713
- Thomas O’Brien and William Reddy, Each compact orientable surface of positive genus admits an expansive homeomorphism, Pacific J. Math. 35 (1970), 737–741. MR 276953
- W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769–774. MR 38022, DOI 10.1090/S0002-9939-1950-0038022-3
- W. R. Utz, Expansive mappings, Topology Proc. 3 (1978), no. 1, 221–226 (1979). MR 540492
- Robert F. Williams, A note on unstable homeomorphisms, Proc. Amer. Math. Soc. 6 (1955), 308–309. MR 68211, DOI 10.1090/S0002-9939-1955-0068211-8
- Richard K. Williams, On expansive homeomorphisms, Amer. Math. Monthly 76 (1969), 176–178. MR 238278, DOI 10.2307/2317269
- Richard K. Williams, Some results on expansive mappings, Proc. Amer. Math. Soc. 26 (1970), 655–663. MR 266186, DOI 10.1090/S0002-9939-1970-0266186-8
- Richard K. Williams, Further results on expansive mappings, Proc. Amer. Math. Soc. 58 (1976), 284–288. MR 436103, DOI 10.1090/S0002-9939-1976-0436103-1
Bibliographic Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 86 (1982), 531-534
- MSC: Primary 54H20; Secondary 54C10
- DOI: https://doi.org/10.1090/S0002-9939-1982-0671230-8
- MathSciNet review: 671230