Planar embeddings of Minc’s continuum and generalizations. (English) Zbl 1460.54010
S. B. Nadler jun. and J. Quinn [Embeddability and structure properties of real curves. Providence, RI: American Mathematical Society (AMS) (1972; Zbl 0236.54029)] asked whether for every point \(x\) of a chainable continuum \(X\) there is a planar embedding of \(X\) for which \(x\) is accessible. One of the candidates to find a counterexample was a continuum \(X_M\) constructed by Minc, see [Lect. Notes Pure Appl. Math. 230, 331–339 (2002; Zbl 1040.54500)], and the main result here gives a positive answer to the question of Nadler and Quinn for a class of continua including \(X_M\). The class of continua studied here are inverse limits \(I\overset{f}{\leftarrow}I\overset{f}{\leftarrow}\cdots\) of intervals with bonding map \(f\) a piecewise monotone post-critically finite locally eventually onto map. For this class it is shown that for every point there is a planar embedding that renders the point accessible.
Reviewer: Thomas B. Ward (Leeds)
MSC:
| 54C25 | Embedding |
| 54F15 | Continua and generalizations |
| 37B45 | Continua theory in dynamics |
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
| 37E05 | Dynamical systems involving maps of the interval |
| 54F17 | Inverse limits of set-valued functions |
References:
| [1] | Anušić, A.; Bruin, H.; Činč, J., Uncountably many planar embeddings of unimodal inverse limit spaces, Discrete Contin. Dyn. Syst., Ser. A, 37, 2285-2300 (2017) · Zbl 1357.37017 |
| [2] | Anušić, A.; Bruin, H.; Činč, J., Problems on planar embeddings of chainable continua and accessibility, (Problems in Continuum Theory in Memory of Sam B. Nadler, Jr. Ed. Logan Hoehn, Piotr Minc, Murat Tuncali. Problems in Continuum Theory in Memory of Sam B. Nadler, Jr. Ed. Logan Hoehn, Piotr Minc, Murat Tuncali, Topology Proc., vol. 52 (2018)), 283-285 |
| [3] | Anušić, A.; Bruin, H.; Činč, J., Planar embeddings of chainable continua, Topol. Proc., 56, 263-296 (2020) · Zbl 1458.37025 |
| [4] | Anušić, A.; Činč, J., Accessible points of planar embeddings of tent inverse limit spaces, Diss. Math., 541 (2019), 57 pp · Zbl 1431.37012 |
| [5] | Barge, M.; Martin, J., Construction of global attractors, Proc. Am. Math. Soc., 110, 523-525 (1990) · Zbl 0714.58036 |
| [6] | Bing, R. H., Snake-like continua, Duke Math. J., 18, 653-663 (1951) · Zbl 0043.16804 |
| [7] | Brechner, B., On stable homeomorphisms and imbeddings of the pseudo-arc, Ill. J. Math., 22, 4, 630-661 (1978) · Zbl 0386.54021 |
| [8] | Brown, M., Some applications of an approximation theorem for inverse limits, Proc. Am. Math. Soc., 11, 478-483 (1960) · Zbl 0113.37705 |
| [9] | Debski, W.; Tymchatyn, E., A note on accessible composants in Knaster continua, Houst. J. Math., 19, 3, 435-442 (1993) · Zbl 0790.54041 |
| [10] | (Hoehn, L.; Minc, P.; Tuncali, M., Problems in Continuum Theory in Memory of Sam B. Nadler, Jr.. Problems in Continuum Theory in Memory of Sam B. Nadler, Jr., Topology Proc., vol. 52 (2018)), 281-308 · Zbl 1391.54001 |
| [11] | (Illanes, A.; Maciás, S.; Lewis, W., Continuum Theory: Proceedings of the Special Session in Honor of Professor Sam B. Nadler, Jr.’s 60th Birthday. Continuum Theory: Proceedings of the Special Session in Honor of Professor Sam B. Nadler, Jr.’s 60th Birthday, Lecture Notes in Pure and Applied Mathematics, vol. 230 (2002), Marcel Dekker: Marcel Dekker New York) · Zbl 1005.00053 |
| [12] | Lewis, W., Embeddings of the pseudo-arc in \(E^2\), Pac. J. Math., 93, 1, 115-120 (1981) · Zbl 0459.57005 |
| [13] | Mahavier, W. S., A chainable continuum not homeomorphic to an inverse limit on \([0, 1]\) with only one bonding map, Proc. Am. Math. Soc., 18, 284-286 (1967) · Zbl 0152.21302 |
| [14] | Mayer, J. C., Embeddings of plane continua and the fixed point property, 1-247 (1982), University of Florida, Ph. D. dissertation |
| [15] | Mayer, J. C., Inequivalent embeddings and prime ends, Topol. Proc., 8, 99-159 (1983) · Zbl 0546.54030 |
| [16] | Mazurkiewicz, S., Un théorème sur l’accessibilité des continus indécomposables, Fundam. Math., 14, 271-276 (1929) · JFM 55.0978.01 |
| [17] | de Melo, W.; van Strien, S., One-Dimensional Dynamics (1993), Springer: Springer Berlin, Heidelberg, New York · Zbl 0791.58003 |
| [18] | Minc, P., Embedding tree-like continua in the plane, (Problems in Continuum Theory in Memory of Sam B. Nadler, Jr. Ed. Logan Hoehn, Piotr Minc, Murat Tuncali. Problems in Continuum Theory in Memory of Sam B. Nadler, Jr. Ed. Logan Hoehn, Piotr Minc, Murat Tuncali, Topology Proc., vol. 52 (2018)), 294-299 |
| [19] | Minc, P.; Transue, W. R.R., Accessible points of hereditarily decomposable chainable continua, Trans. Am. Math. Soc., 2, 711-727 (1992) · Zbl 0752.54012 |
| [20] | Mioduszewski, J., Mappings of inverse limits, Colloq. Math., 10, 39-44 (1963) · Zbl 0118.18205 |
| [21] | Nadler, S. B., Continuum Theory: An Introduction (1992), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York · Zbl 0757.54009 |
| [22] | Nadler, S. B., Some results and problems about embedding certain compactifications, (Proceedings of the University of Oklahoma Topology Conference (1972)), 222-233 · Zbl 0283.54017 |
| [23] | Nadler, S. B.; Quinn, J., Embeddability and Structure Properties of Real Curves, Memoirs of the American Mathematical Society, vol. 125 (1972), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0236.54029 |
| [24] | Ozbolt, J., On Planar Embeddings of the Knaster VΛ-Continuum, 1-51 (2020), Auburn University, PhD dissertation |
| [25] | Smith, M., Plane indecomposable continua no composant of which is accessible at more than one point, Fundam. Math., 111, 61-69 (1981) · Zbl 0458.54024 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
