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Universal dendrites of order m$

Let m \in
\{3,4,\dots,\omega\}$. By the standard universal dendrite of order m$ we mean a dendrite D_m$ such that each ramification point of D_m$ is of order m$ and for each arc A \subset D_m$ the set of ramification points of D_m$ which belong to A$ is dense in A$. Their constructions, known again from [Wazewski 1923, Chapter K, p. 137] (see also [Charatonik 1991, (4), p. 168]; for the inverse limit construction see [Chaaratonik 1980, p. 491]), mimic that of the Wazewski universal dendrite D_\omega $, but instead of copies of F_\omega $ we use copies of m$-ods at each step of the construction. See Figures A-C for the standard universal dendrites D_3$, D_4$ and D_6$.

Figure 1.3.7: ( A ) standard universal dendrite D_3$

Figure: ( AA ) standard universal dendrite D_3$ - an animation

Figure 1.3.7: ( B ) standard universal dendrite D_4$

Figure: ( BB ) standard universal dendrite D_4$ - an animation

Figure: ( BBB ) standard universal dendrite D_4$ produced as an intersection - an animation

Figure 1.3.7: ( C ) standard universal dendrite D_6$

The standard universal dendrites D_m$ have the following properties.

  1. For each m \in
\{3,4,\dots,\omega\}$ D_m$ is universal in the class of all dendrites for which the order of their ramification points is less than or equal to m$, see e.g. [Menger 1932, Chapter 10, § 6, p. 322].
  2. If m, n \in \mathbb{N}$ with 3 \le m < n$, then there exists an open mapping of D_n$ onto D_m$, [Chaaratonik 1980, Theorem 2, p. 492].
  3. Among all standard universal dendrites D_m$ only D_3$ and D_\omega $ are homeomorphic with all their open images, [Chaaratonik 1980, Corollary, p. 493].
  4. For each m \in
\{3,4,\dots,\omega\}$ a monotone surjection of D_m$ onto itself is a near homeomorphism if and only if m = 3$, [Charatonik 1991, Corollary 5.5, p. 178].
  5. Any two standard universal dendrites D_m$ and D_n$ of some orders m, n \in \{3, 4, \dots, \omega\}$ are monotonely equivalent, [Charatonik 1991, Corollary 6.6, p. 180].
  6. For each m \in
\{3,4,\dots,\omega\}$ the dendrite D_m$ is monotonely homogeneous [Charatonik 1991, Theorem 7.1, p. 186].

Other mapping properties of the standard universal dendrites D_m$ can be found e.g. in [Chaaratonik 1980], [Charatonik 1991], [Charatonik 1995], [Charatonik et al. 1997a], [Charatonik et al. 1998], [Charatonik et al. 1994] and [Charatonik W.J. et al. 1994].

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next up previous contents index
Next: Other universal dendrites Up: Dendrites Previous: Wazewski universal dendrite
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih