On the operator of center of distances between the spaces of closed subsets of the real line
DOI:
https://doi.org/10.12775/TMNA.2023.023Słowa kluczowe
Center of distances, Hausdorff metric, semicontinuity, orbit, CantorvalAbstrakt
We study properties of an operator $S$ which assigns to compact subsets of $[0,1]$ their centers of distances. We consider its continuity points and its upper semicontinuity points as well as orbits and fixed points of this operator. We also compute centers of distances of some classic sets. Using properties of operator $S$ we show that the family of achievement sets is of the first category in the space of compact subsets of $[0,1]$.Bibliografia
M.F. Barnsley, Fractals Everywhere, Academic Press, 2nd edition, 1993.
T. Banakh, A. Bartoszewicz, M. Filipczak and E. Szymonik, Topological and measure properties of some self-similar sets, Topol. Methods Nonlinear Anal. 46 (2015), 1013–1028.
M. Banakiewicz, The center of distances of central Cantor sets, Results Math. 78 (2023), 234.
M. Banakiewicz, A. Bartoszewicz, M. Filipczak and F. Prus-Wiśniowski, Center of distances and central Canor sets, Results Math. 77 (2022), article number 196.
A. Bartoszewicz, M. Filipczak and F.Prus-Wiśniowski, Topological and algebraic aspects of subsums of series, Traditional and Present-day Topics in Real Analysis, Faculty of Mathematics and Computer Science, University of Łódź, Łódź, 2013, pp. 345–366.
A. Bartoszewicz, S. Głąb and J. Marchwicki, Recovering purely atomic finite measure from its range, J. Math. Anal. Appl. 467 (2018), no. 2, 825–841.
A. Bartoszewicz, S. Głąb, M. Filipczak, F. Prus-Wiśniowski and J. Swaczyna, On generating regular Cantorvals connected with geometric Cantor sets, Chaos Solitons Fractals 114, (2018), 468–473.
W. Bielas, S. Plewik and M. Walczyńska, On the center of distances, Eur. J. Math. 4 (2018), 687–698.
K. Drakakis, A review of the available construction methods for Golomb rulers, Adv. Math. Commun. 3 (2009), no. 3, 235–250.
J.A. Guthrie and J.E. Nymann, The topological structure of the set of subsums of an infinite series, Colloq. Math. 55 (1988), no. 2, 323–327.
R. Jones, Achievement sets of sequences, Amer. Math. Monthly 118 (2011), no. 6, 508–521.
M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, second ed., Birkhäuser Verlag, Basel, 2009.
M. Kula and P. Nowakowski, Achievement sets of series in R2 (2023), arXiv: 2312.08155[math.CA].
K. Kuratowski, Topology, vol. II, Academic Press, 1968.
P. Mendes and F. Oliveira, On the topological structure of the arithmetic sum of two Cantor sets, Nonlinearity 7 (1994), 329–343.
Z. Nitecki, Cantorvals and Subsum Sets of Null Sequences, Amer. Math. Monthly 122 (2015), no. 9, 862–870.
J.E. Nymann and R.A. Sáenz, On the paper of Guthrie and Nymann on subsums of infinite series, Colloq. Math. 83 (2000), 1–4.
C.A. Rogers, Hausdorff Measures, Cambridge University Press, 1970.
B. Santiago, The semicontinuity lemma (2012), preprint: http://www.professores.uff.br/brunosantiago/wp-con tent/uploads/sites/17/2017/07/01.pd.
Pobrania
Opublikowane
Jak cytować
Numer
Dział
Licencja
Prawa autorskie (c) 2023 Artur Bartoszewicz, Małgorzata Filipczak, Grażyna Horbaczewska, Sebastian Lindner, Franciszek Prus-Wiśniowski
Utwór dostępny jest na licencji Creative Commons Uznanie autorstwa – Bez utworów zależnych 4.0 Międzynarodowe.
Statystyki
Liczba wyświetleń i pobrań: 0
Liczba cytowań: 0