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 having the set homeomorphic to the Cantor ternary set in such that all ramification points of 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

The Gehman dendrite has the following properties.

1. The set of all ramification points of is discrete.
2. .
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 contains the Gehman dendrite if and only if does not have the periodic-recurrent property, [Illanes 1998, Theorem 2, p. 222].

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