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Gehman dendrite

One of the classical examples of dendrites with its set of end points closed is the Gehman dendrite . It can be described as a dendrite G$ having the set E(G)$ homeomorphic to the Cantor ternary set C$ in [0,1]$ such that all ramification points of G$ are of order 3 (see [Gehman 1925, the example on p. 42]; see also [Nikiel 1983, p. 422-423] for a geometrical description; compare [Nikiel 1989, p. 82] and [Nadler 1992, Example 10.39, p. 186]). See Figure A.

Figure 1.3.18: ( A ) Gehman dendrite
A.gif

The Gehman dendrite has the following properties.

  1. The set R(G)$ of all ramification points of G$ is discrete.
  2. E(G) = \mathrm{cl}\,(R(G)) \setminus R(G)$.
  3. Each dendrite with an uncountable set of its end points contains a homeomorphic copy of the Gehman dendrite, [Arévalo et al. 2001, Proposition 6.8, p. 16].

  4. If a continuum contains the Gehman dendrite, then it does not have the periodic-recurrent property, [Charatonik 1998, Theorem 3.3, p. 136].
  5. A dendrite X$ contains the Gehman dendrite if and only if X$ does not have the periodic-recurrent property, [Illanes 1998, Theorem 2, p. 222].

Here you can find source files of this example.

Here you can check the table of properties of individual continua.

Here you can read Notes or write to Notes ies of individual continua.
next up previous contents index
Next: Modifications of the Gehman Up: Dendrites Previous: Dendrites with the closed
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30