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We revisit the results from two papers, Mioduszewski’s “Mappings of inverse limits” and Feuerbacher’s “Mappings of inverse limits revisited” to obtain new mapping theorems for inverse limits of inverse sequences of compact metric spaces with continuous single-valued bonding functions. Then, we apply the results to the theory of inverse limits of inverse sequences of compact metric spaces with upper semicontinuous setvalued bonding functions to obtain new mapping theorems for such inverse limits. This answers an open problem stated by Ingram.

We consider a family of inverse limits of inverse sequences of closed unit intervals with a single upper semi-continuous set-valued bonding function whose graph is an arc ; the graph is the union of two line segments in [0, 1]^2, both of which contain the origin (0, 0) and have positive slope. One of the segments extends to the top-boundary and the other to the right side boundary of [0, 1]×[0, 1]. We show that there is a large subfamily F of these bonding functions such that for each f ∈ F, the inverse limit of the inverse sequence of closed unit intervals using f as a single bonding function is homeomorphic to the Lelek fan.

The paper is motivated by E. Akin's book about dynamical systems and closed relations [1], and by J. Kennedy's and G. Erceg's recent paper about the entropy of closed relations on closed intervals [9].In present paper, we introduce the entropy of a closed relation \begin{document}$ G $\end{document} on any compact metric space \begin{document}$ X $\end{document} and show its basic properties. We also introduce when such a relation \begin{document}$ G $\end{document} generates a periodic point or finitely generates a Cantor set. Then we show that periodic points, finitely generated Cantor sets, Mahavier products and the entropy of closed relations are preserved by topological conjugations. Among other things, this generalizes the well-known results about the topological conjugacy of continuous mappings. Finally, we prove a theorem, giving sufficient conditions for a closed relation \begin{document}$ G $\end{document} on \begin{document}$ [0,1] $\end{document} to have a non-zero entropy. Then we present various examples of closed relations \begin{document}$ G $\end{document} on \begin{document}$ [0,1] $\end{document} such that (1) the entropy of \begin{document}$ G $\end{document} is non-zero, (2) no periodic point or exactly one periodic point is generated by \begin{document}$ G $\end{document}, and (3) no Cantor set is finitely generated by \begin{document}$ G $\end{document}.

Given compact metric spaces X 1 and X 2 , closed subsets Y 1 and Y 2 of X 1 and X 2 respectively, and an upper semicontinuous function g : Y 1 ⊸ Y 2 , we define a special kind of extension f : X 1 ⊸ X 2 of the function g and prove some of its properties. Using this special extension, we answer a question stated by Banic, Crepnjak, Merhar and Milutinovic. We also introduce the notions of the weak L -extension property and the strong L -extension property of an inverse sequence for any property L of graphs of set-valued functions. We show that, in the case where L means the surjectivity of such graphs, not all inverse sequences of compact metric spaces have the strong L -extension property but they do have the weak L -extension property.

We give several characterizations of inverse limits of compact metric spaces with upper semicontinuous set-valued bonding functions having the property that any closed subset of the inverse limit is the inverse limit of its projections. This solves a problem stated by Ingram.

In the paper, we give new results about dimension of inverse limits with upper semicontinuous bonding functions. We also construct an upper semicontinuous function f : [ 0 , 1 ] → 2 [ 0 , 1 ] with the following properties: (1) the graph of f is an arc, (2) the graph of f is surjective, (3) the inverse limit lim ← { [ 0 , 1 ] , f } n = 1 ∞ contains a simple closed curve, and (4) the dimension of lim ← { [ 0 , 1 ] , f } n = 1 ∞ is 1. This answers a question by W.T. Ingram.

In 2002 Markov interval maps were introduced by S. Holte. It was shown that any two inverse limits with Markov interval bonding maps with the same pattern were homeomorphic. In this article we introduce generalized Markov interval functions, which are a generalization of Markov interval maps to set-valued functions, and show that any two generalized inverse limits with generalized Markov interval bonding functions with the same pattern are homeomorphic.

In this paper several characterizations of subspaces of products of countably many compact metric spaces that are inverse limits of inverse sequences are given. These characterizations are obtained both for inverse sequences with continuous single-valued bonding functions and for inverse sequences with upper semicontinuous set-valued bonding functions. The proofs are based on use of crossovers and inverse limit hulls.

We introduce an apparatus to measure closeness or relationship of two given objects. This is a topology-based apparatus that uses graph representations of the compared objects. In more detail, we obtain a metric on the class of all pairwise non-isomorphic connected simple graphs to measure closeness of two such graphs. To obtain such a measure, we use the theory of hyperspaces from topology to introduce the notion of the Hausdorff graph \(2^G \) of any graph G. Then, using this new concept of Hausdorff graphs combined with the notion of graph amalgams, we present the Hausdorff distance, which proves to be useful when examining the closeness of any two connected simple graphs. We also present many possible applications of these concepts in various areas.

We give new results about the set of all medians, the set of all first quartiles and the set of all third quartiles of a finite dataset. We also give new and interesting results about rela- tionships between these sets. We also use these results to provide an elementary correctness proof of the Langford's doubling method.

In the theory of generalized inverse limits it is a well-known fact that the generalized inverse limits may not be connected even if all the factor spaces are closed intervals. However, it has been shown recently by Banic and Kennedy that such generalized inverse limits always contain large continua, if the bonding functions have connected and surjective graphs. We generalize the notion of generalized inverse limits of inverse sequences of closed intervals with upper semicontinuous bonding functions to inverse limits of inverse systems over directed graphs and show that under certain conditions, such inverse limits also contain large continua.

We give new results about the full projection property and the weak full projection property for inverse limits with upper semicontinuous bonding functions. This solves several problems stated by W. T. Ingram.

We give a characterization of inverse sequences with upper semicontinuous bonding functions fi:[0,1] ⊸ [0,1] for which the inverse limit of the inverse sequence with fi-1 as bonding functions is connected. As a byproduct, we obtain another characterization of connected inverse limits of inverse sequences with a single bonding function.

In this paper, we explore connectedness and total disconnectedness of generalized inverse limits on intervals induced by one bonding function f : I → 2 I , which is not necessarily surjective, and whose graph G ( f ) is an arc. As the main results of the paper we prove that (1) if such inverse limit consists of more than one point, then it is infinite, and (2) if such a function f is surjective, then the corresponding inverse limit is never totally disconnected. As a by-product, we obtain a more general result: if for each i, f i denotes a surjective upper semicontinuous bonding function with connected graph, then the resulting generalized inverse limit is never totally disconnected. We also produce examples of such inverse limits (with nonsurjective bonding function f) that are totally disconnected.

We investigate inverse limits in the category $ \mathcal{CHU} $ of compact Hausdorff spaces with upper semicontinuous functions. We introduce the notion of weak inverse limits in this category and show that the inverse limits with upper semicontinuous set-valued bonding functions (as they were defined by Ingram and Mahavier [‘Inverse limits of upper semi-continuous set valued functions’, Houston J. Math. 32 (2006), 119–130]) together with the projections are not necessarily inverse limits in $ \mathcal{CHU} $ but they are always weak inverse limits in this category. This is a realisation of our categorical approach to solving a problem stated by Ingram [An Introduction to Inverse Limits with Set-Valued Functions (Springer, New York, 2012)].

We study generalized tent maps inverse limits, i.e. inverse limits of inverse sequences of unit segments I with a generalized tent map being the only bonding function. As the main result we identify an infinite family of curves in I 2 such that if top points of graphs of generalized tent maps belong to the same curve, the corresponding inverse limits are homeomorphic, and if they belong to different curves, the inverse limits are non-homeomorphic. The inverse limits corresponding to certain families of top points are explicitly determined, and certain properties of the inverse limit are proved in the case of ( 0 , 1 ) as the top point.

In Banic, Crepnjak, Merhar and Milutinovic (2010) [2] the authors proved that if a sequence of graphs of surjective upper semi-continuous set-valued functions f n : X → 2 X converges to the graph of a continuous single-valued function f : X → X , then the sequence of corresponding inverse limits obtained from f n converges to the inverse limit obtained from f. In this paper a more general result is presented in which surjectivity of f n is not required. The result is also generalized to the case of inverse sequences with non-constant sequences of bonding maps. Finally, these new theorems are applied to inverse limits with tent maps. Among other applications, it is shown that the inverse limits appearing in the Ingram conjecture (with a point added) form an arc.

Fault tolerance and transmission delay of networks are important concepts in network design. The notions are strongly related to connectivity and diameter of a graph, and have been studied by many authors. Wide diameter of a graph combines studying connectivity with the diameter of a graph. Diameter with width k of a graph G, k-diameter, is defined as the minimum integer d for which there exist at least k internally disjoint paths of length at most d between any two distinct vertices in G. Denote by Dc(G) the c-diameter of G and κ(G) the connectivity of G. In the context of computer networks, wide diameters of Cartesian graph products have been recently studied by many authors. Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let G be a Cartesian graph bundle with fiber F over base B, 0

We study the following problem: if a sequence of graphs of upper semi-continuous set valued functions fn converges to the graph of a function f, is it true that the sequence of corresponding inverse limits obtained from fn converges to the inverse limit obtained from f?

Let DqE(G) denote the maximum diameter among all subgraphs obtained by deleting q edges of G. Let DpV(G) denote the maximum diameter among all subgraphs obtained by deleting p vertices of G. We prove that DaE(G)⩽DaV(G)+1 for all meaningful a. We also define mixed fault diameter D(p,q)M(G), where p vertices and q edges are deleted at the same time. We prove that for 0

We show that two inverse limits of inverse sequences of closed intervals and quasi Markov bonding functions are homeomorphic, if the inverse sequences follow the same pattern. This significantly improves Holte's result about when two inverse limits of inverse sequences with Markov interval maps as bonding functions are homeomorphic. Our result improves Holte's result in several directions: (1) it generalizes finite Markov partitions to Markov pairs of sets that may even be uncountable, (2) it generalizes Markov interval maps $I\rightarrow I$ to quasi Markov functions $I\rightarrow J$ (so the domains and the codomains of the bonding functions are not necessarily the same interval), (3) we no longer require the bonding functions to be surjective, and (4) we no longer require the bonding functions to be continuous.

Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let G be a kG-connected graph and Dc(G) denote the diameter of G after deleting any of its c < kG vertices. We prove that Da+b+1(G) ≤ Da(F) + Db(B) + 1 if G is a graph bundle with fibre F over base B, a < kF, and b < kB.

In calculus, an indefinite integral of a function f is a differentiable function F whose derivative is equal to f. The main goal of the paper is to generalize this notion of the indefinite integral from the ring of real functions to any ring. We also investigate basic properties of such generalized integrals and compare them to the well-known properties of indefinite integrals of real functions.

We construct a family of upper semi-continuous set-valued functions f:[0,1] → 2[0,1] (belonging to the class of so-called comb functions), such that for each of them the inverse limit of the inverse sequence of intervals [0,1] and f as the only bonding function is homeomorphic to Ważewski's universal dendrite. Among other results we also present a complete characterization of comb functions for which the inverse limits of the above type are dendrites.

A Cartesian graph bundle is a generalization of a graph covering and a Cartesian graph product. Let G be a kG-edge connected graph and D̄c(G) be the largest diameter of subgraphs of G obtained by deleting c

Let G be a k-connected graph and Dc(G) denote the maximum diameter of G after deleting any of its c

We show that the inverse limit of any inverse sequence of compact metric spaces and surjective bonding maps is in fact the limit of a sequence of homeomorphic copies of the same spaces with respect to the Hausdorff metric.