Finite Graphs and Inverse Limits with Set-Valued Functions on Intervals
Keywords:
finite graph, inverse limit with set-valued functions, Mahavier productAbstract
If the inverse limit of upper semi-continuous set-valued functions from $[0, 1]$ to the closed subsets of $[0, 1]$ is a finite graph $G$, then for some $N$ and all $n \geq N$, the projection onto the first $n$ coordinates of that inverse limit is a projection onto a finite graph that is homeomorphic to $G$.
References
John Philip Huneke, Mountain climbing, Trans. Amer. Math. Soc. 139 (1969), 383-391.
W. T. Ingram, Concerning chainability of inverse limits on [0,1] with set-valued functions, Topology Proc. 42 (2013), 327-340.
Judy Kennedy and Van Nall, Dynamical properties of shift maps on inverse limits with a set valued function, Ergodic Theory Dynam. Systems 38 (2018), no. 4, 1499-1524.
Verónica Martínez-de-la-Vega and Ivon Vidal-Escobar, Dendrites on generalized inverse limits and finite Mahavier products, Topology Appl. 222 (2017), 238-253.
J. Mioduszewski, On a quasi-ordering in the class of continuous mappings of a closed interval into itself, Colloq. Math. 9 (1962), 233-240.
Van C. Nall, Maps which preserve graphs, Proc. Amer. Math. Soc. 101 (1987), no. 3, 563-570.
Van Nall, The only finite graph that is an inverse limit with a set valued function on [0,1] is an arc, Topology Appl. 159 (2012), no. 3, 733-736.
R. H. Sorgenfrey, Concerning triodic continua, Amer. J. Math. 66 (1944), 439-460.
Ivon Vidal-Escobar, Properties of the shift map on dendrites that are generalized inverse limits, Houston J. Math. 45 (2019), no. 1, 1-19.
