arXiv subjects
Publications and source records attributed to Judy Kennedy.
Among other results, the paper gives new mapping theorems and new fixed point property theorems for inverse limits of inverse sequences of compact metric spaces with upper semicontinuous set-valued bonding functions. We also revisit the results from two papers, Mioduszewski's ''Mappings of inverse limits'' and Feuerbacher's ''Mappings of inverse limits revisited''.
We give two characterizations of $\mathcal P$-like continua $X$ that do not have the fixed point property. Both characterizations are stated in terms of sequences of open covers of $X$ that follow fixed-point-free patterns. We use these to characterize planar tree-like continua that do not have the fixed point property in terms of infinite sequences of tree-chains in the plane that follow fixed-point-free patterns. We also establish a useful relationship between these tree-chains and commutative simplicial diagrams that we use later to construct a finite sequence (of any given length) of tree-chains in the plane that follows a fixed-point-free pattern. An earlier characterization of $\mathcal P$-like continua with the fixed point property was given in 1994 by Feuerbacher based on a 1963 result by Mioduszewski. The Mioduszewski-Feuerbacher characterization is expressed in terms of almost commutative inverse diagrams. In contrast, our approach is more geometric, and it may potentially lead to new methods in the elusive search for a planar tree-like continuum without the fixed-point property.
In this article, we show that R.H. Bing's pseudo-circle admits a minimal non-invertible map. This resolves a problem raised by Bruin, Kolyada and Snoha in the negative. The main tool is the Denjoy-Rees technique, further developed by Béguin-Crovisier-Le Roux, combined with detailed study into the structure of the pseudo-circle.
We generalize the definition of topological entropy due to Adler, Konheim, and McAndrew \cite{AKM} to set-valued functions from a closed subset $A$ of the interval to closed subsets of the interval. We view these set-valued functions, via their graphs, as closed subsets of $[0,1]^{2}$. We show that many of the topological entropy properties of continuous functions of a compact topological space to itself hold in our new setting, but not all. We also compute the topological entropy of some examples, relate the entropy to other dynamical and topological properties of the examples, and we give an example of a closed subset $G$ of $[0,1]^2$ that has $0$ entropy but $G\cup \{ (p,q) \},$ where $(p,q) \in [0,1]^2\setminus G,$ has infinite entropy.