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2. Continua with kernelsIztok Banič, 2008, original scientific article Abstract: In this article we introduce the concept of kernels of continua, obtained by combining inverse limits of inverse sequences of unit intervals and one-valued bonding maps with inverse limits of inverse sequences of unit intervals and upper semicontinuous set-valued bonding functions. We also show some of their properties, with special emphasis on arc-like continua. Keywords: mathematics, kernels, arc-like continua Published: 01.06.2012; Views: 799; Downloads: 25 Link to full text |
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4. Tent maps inverse limits and open problemsMatevž Črepnjak, Iztok Banič, Matej Merhar, Uroš Milutinović, 2011, published scientific conference contribution abstract Keywords: mathematics, topology, continua, inverse limits, tent maps, Knaster continua Published: 07.06.2012; Views: 696; Downloads: 25 Link to full text |
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6. The Wazewski's univerzal dendrite as an inverse limit with single bonding functionTina Sovič, Iztok Banič, Matevž Črepnjak, Matej Merhar, Uroš Milutinović, 2011, published scientific conference contribution abstract Keywords: mathematics, topology, continua, inverse limits, dendrites Published: 07.06.2012; Views: 886; Downloads: 24 Link to full text |
7. Paths through inverse limitsIztok Banič, Matevž Črepnjak, Matej Merhar, Uroš Milutinović, 2011, original scientific article Abstract: In Banič, Črepnjak, Merhar and Milutinović (2010) [2] the authors proved that if a sequence of graphs of surjective upper semi-continuous set-valued functions ▫$f_n : X to 2^X$▫ converges to the graph of a continuous single-valued function ▫$f : X to X$▫, 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. Keywords: mathematics, topology, continua, limits, inverse limits, upper semi-continuous set-valued functions, paths, arcs Published: 07.06.2012; Views: 915; Downloads: 62 Link to full text |
8. Two constructions of continua: inverse limits and compactificationsTina Sovič, 2013, doctoral dissertation Abstract: In the thesis we talk about two different constructions of continua. First we present the generalized inverse limits, with help of which we construct Wazewski's universal dendrite. What follows is a description of the compactifications of a ray and the presentation of results about their span.
The first chapter will be an introduction to the continuum theory trough interesting examples, as sin(1/x)-continuum, Hilbert cube, Brouwer-Janiszewski-Knaster continuum and pseudoarc. We will present some of their properties, among which irreducibility, smoothness and span zero are the most important ones for us.
In the continuation we intend to present some various constructions of continua. The main focus will be on the generalized inverse limits and compactifications of rays, which will also be a central part of the thesis. In this chapter, we also study inverse limits in the category of compact Hausdorff spaces with upper semi-continuous functions. We show that the inverse limits with upper semi-continuous bonding functions, together with the projections are weak inverse limits in this category.
The following two are the most important chapters in the thesis. The first is a detailed description of a construction of the family of upper semi-continuous functions f, such that the inverse limit of the inverse sequence of unit intervals and f, as the only bonding function, is homeomorphic to Wazewski's universal dendrite for each of it. Among other results we will also give a complete characterization of comb-functions, for which the inverse limits of the type described above are dendrites.
The next important chapter will be about compactifications of rays. In the first part of this chapter we will use compactifications to prove that for each continuum Y there is an irreducible smooth continuum that contains a topological copy of Y. The second part presents the main results of this chapter; i.e. the span of a compactification of a ray with a remainder that has a span zero is also zero. In the proofs of this chapter we will help ourselves with a discretization of span. Keywords: continua, inverse limit, inverse sequence, upper semi-continuous function, set-valued functions, bonding function, hyperspace, dendrite, universal dendrite, category, compactification, compactification of a ray, smooth continua, irreducible continua, span, span zero Published: 25.09.2013; Views: 1409; Downloads: 91 Full text (861,84 KB) |
9. Paths through inverse limitsIztok Banič, Matevž Črepnjak, Matej Merhar, Uroš Milutinović, 2009 Abstract: In [I.Banič, M. Črepnjak, M. Merhar, U. Milutinović, Limits of inverse limits, Topology Appl. 157 (2010) 439-450] the authors proved that if a sequence of graphs of surjective upper semi-continuous set-valued functions ▫$f_n: X rightarrow 2^X$▫ converges to the graph of a continuous single-valued function ▫$f: X rightarrow X$▫, 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. Also, the result is 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. Keywords: matematika, topologija, kontinuumi, limite, inverzne limite, navzgor polzvezne večlične funkcije, poti, loki, mathematics, topology, continua, limits, inverse limits, upper semi-continuous set-valued functions, paths, arcs Published: 10.07.2015; Views: 589; Downloads: 42 Link to full text |
10. Towards the complete classification of tent maps inverse limitsIztok Banič, Matevž Črepnjak, Matej Merhar, Uroš Milutinović, 2010 Abstract: We study tent map inverse limits, i.e. inverse limits of inverse sequences of unit segments ▫$I$▫ with a 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 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. Keywords: matematika, topologija, kontinuumi, inverzne limite, mathematics, topology, continua, inverse limits, tent maps, Knaster continua Published: 10.07.2015; Views: 425; Downloads: 16 Link to full text |