Dendrites on generalized inverse limits and finite Mahavier products (Q524349)

The authors study the relationship between generalized inverse limits and the finite Mahavier product. More precisely, if \(f : [0,1] \to 2^{[0,1]}\) is an upper semicontinuous function and \(M= \{(x,y) \in [0,1]^2 \mid y \in f(x) \}\) is its graph, then the generalized inverse limit is defined by \[ \displaystyle\lim_{\longleftarrow} M = \displaystyle\lim_{\longleftarrow}\{f, [0,1]\}_{n=1}^\infty= \{(x_1, x_2, x_3, \ldots ) \in \prod_{n=1}^\infty [0,1] \;| \;\forall i \in {\mathbb N}: (x_{i+1},x_i) \in M) \}, \] and the finite Mahavier product is defined by, for each positive integer \(k\), \[ G_k = \{(x_1, x_2, x_3, \ldots , x_{k+1}) \in \prod_{n=1}^{k+1} [0,1] \mid \forall i \in \{1,2,\ldots,k\}: (x_{i+1},x_i) \in M) \}. \] From these two definitions, one can see that these two concepts are very close, but we still do not have many results which explain the relation between them, so every such result is a very important contribution to this theory. In the second section, the authors define coincidence sets. More precisely, if \(u,v \in [0,1]\), \(u\leq v\), and \(f,h: [0,1] \to [u,v]\) are maps, then \(E(f,h) = \{(x,y) \in [0,1]^2 \;| \;f(x) = h(y) \}\) is the coincidence set. Note that the authors use this tool instead of the Mountain climbing theorem, cf. \textit{J. P. Huneke} [Trans. Am. Math. Soc. 139, 383--391 (1969; Zbl 0175.34503)]. By using coincidence sets they generalize some results of \textit{J. Mioduszewski} [Colloq. Math. 9, 233--240 (1962; Zbl 0107.27603)], and \textit{R. Sikorski} and \textit{K. Zarankiewicz} [Fundam. Math. 41, 339--344 (1955; Zbl 0064.05501)]. In the same section, one can find the result saying that if there exists an arc \(C \in E(f,h)\) such that \((0,0) \in C\) but it is not an end point of \(C\), then \(E(f,h)\) contains a two cell (for more details see Theorem 2.7). The main results of the paper are included in the last two sections. Most of the results contain the assumption that \(M\) is a compact subset of \([0,1]^2\) with the property \(\rho_1(M) = \rho_2(M) = [0,1]\), where \(\rho_1\) and \(\rho_2\) are the corresponding projections. From the many results I want to emphasize the following ones (all from the last section of the paper): {\parindent=0.7cm\begin{itemize}\item[(1)] if \(\displaystyle\lim_{\longleftarrow} M\) is a dendrite, then \(M\) is a dendrite and for every positive integer \(k\), \(G_k\) is a dendrite (Theorem 5.7); \item[(2)] if \(\displaystyle\lim_{\longleftarrow} M\) is a dendrite with a finite set of ramification points, then \(M\) is a dendrite with a finite set of ramification points and for every positive integer \(k\), \(G_k\) is a dendrite with a finite set of ramification points (Corollary 5.8); \item[(3)] if \(\displaystyle\lim_{\longleftarrow} M\) is an arc, then \(M\) is an arc and for every positive integer \(k\), \(G_k\) is an arc (Corollary 5.8). \end{itemize}} The paper is technical, but the authors are very careful about all the details and take great care about clear representation of all concepts.