Set-valued functions from an interval into the closed subsets of an interval arise in various areas of science and mathematical modeling. Research has shown that the dynamics of a single-valued function on a compact space are closely linked to the dynamics of the shift map on the inverse limit with the function as the sole bonding map. For example, it has been shown that with Devaney’s definition of chaos the bonding function is chaotic if and only if the shift map is chaotic. One reason for caring about this connection is that the shift map is a homeomorphism on the inverse limit, and therefore the topological structure of the inverse-limit space must reflect in its richness the dynamics of the shift map. In the set-valued case there may not be a natural definition for chaos since there is not a single well-defined orbit for each point. However, the shift map is a continuous single-valued function so it together with the inverse-limit space form a dynamical system which can be chaotic in any of the usual senses. For the set-valued case we demonstrate with theorems and examples rich topological structure in the inverse limit when the shift map is chaotic (on certain invariant sets). We then connect that chaos to a property of the set-valued function that is a natural generalization of an important chaos producing property of continuous functions.

Abstract. In 1940, O. G. Harrold showed that no arc can be the exactly 2-to-1 continuous image of a metric continuum, and in 1947 W. H. Gottschalk showed that no dendrite is a 2-to-1 image. In 2003 we show that no arc-connected treelike continuum is the 2-to-1 image of a continuum.

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Abstract. A bottleneck in a dendroid is a continuum that intersects every arc connecting two non-empty open sets. Piotr Minc proved that every dendroid contains a point, which we call a center, contained in arbitrarily small bottlenecks. We study the effect that the set of centers in a dendroid has on its structure. We find that the set of centers is arc connected, that a dendroid with only one center has uncountably many arc components in the complement of the center, and that, in this case, every open set intersects uncountably many of these arc components. Moreover, we find that a map from one dendroid to another preserves the center structure if each point inverse has at most countably many components.1. Introduction. The point p is a center of the dendroid D if there are two points c and d in D such that for every ε > 0 there is a continuum C containing p of diameter less than ε and there are two open sets, U containing c and V containing d, such that C intersects every arc from U to V . The concept of a dendroid center was introduced by Piotr Minc and he proved that every dendroid contains at least one center [5, Theorem 3.6]. We used the results of Minc in [1] to show that no continuum maps exactly 2-to-1 onto a dendroid. Minc called the continuum C a bottleneck, the points c and d basin points, and the two open sets U and V basins.We can put all dendroids into distinct and useful classes with strong structural elements determined by the behavior of the centers or the only center. We show that the set of centers of a dendroid is arc connected and a dendroid with more than one center has uncountably many strong centers. We find that every non-endpoint is a strong center in a locally connected dendroid (also called a dendrite), and that the strong centers are dense in a Suslinean dendroid. On the other hand, some dendroids have only one center. For example, the cone point in the cone over a Cantor set is the only center and it is a strong center. A significantly more complicated dendroid