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Semiflows induced by length metrics: on the way to extinction. (English) Zbl 1356.37029
Summary: R. H. Bing [Bull. Am. Math. Soc. 55, 1101–1110 (1949; Zbl 0036.11702)] and E. E. Moise [Bull. Am. Math. Soc. 55, 1111–1121 (1949; Zbl 0036.11801)] proved, independently, that any Peano continuum admits a length metric \(d\). We treat non-degenerate Peano continua with a length metric as evolution systems. For any compact length space \((X, d)\) we consider a semiflow in the hyperspace \(2^X\) of all non-empty closed sets in \(X\). This semiflow starts with a canonical copy of the Peano continuum \((X, d)\) at \(t = 0\) and, at some time, collapses everything into a point. We study some properties of this semiflow for several classes of spaces, manifolds, graphs and finite polyhedra among them.

MSC:
37B45 Continua theory in dynamics
54E35 Metric spaces, metrizability
54B20 Hyperspaces in general topology
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[1] Bhatia, N. P.; Szegö, G. P., Stability theory of dynamical systems, (1970), Springer-Verlag New York · Zbl 0213.10904
[2] Bing, R. H., Partitioning a set, Bull. Am. Math. Soc., 55, 1101-1110, (1949) · Zbl 0036.11702
[3] Bridson, M.; Haefliger, A., Metric spaces of non-positive curvature, (1999), Springer-Verlag Berlin · Zbl 0988.53001
[4] Burago, D.; Burago, Y.; Ivanov, S., A course in metric geometry, Grad. Stud. Math., vol. 33, (2001), AMS Providence, RI · Zbl 0981.51016
[5] Chiswell, I., Introduction to λ-trees, (2001), World Scientific Singapore · Zbl 1004.20014
[6] Curtis, D. W.; Schori, R. M., Hyperspaces of Peano continua are Hilbert cubes, Fundam. Math., 101, 1, 19-38, (1978) · Zbl 0409.54044
[7] Gromov, M., Hyperbolic groups, (Gersten, S. M., Essays in Group Theory, Math. Sci. Res. Inst. Publ., vol. 8, (1987), Springer New York), 75-263 · Zbl 0634.20015
[8] Hatcher, A., Algebraic topology, (2002), Cambridge University Press · Zbl 1044.55001
[9] Hopf, H.; Rinow, W., Über den begriff der vollständingen differentialegeometrischen flächen, Comment. Math. Helv., 3, 209-225, (1931) · JFM 57.0871.04
[10] Hughes, B., Trees and ultrametric spaces: a categorical equivalence, Adv. Math., 189, 148-191, (2004) · Zbl 1061.57021
[11] Hughes, B.; Martinez-Perez, B.; Morón, M. A., Bounded distortion homeomorphisms on ultrametric spaces, Ann. Acad. Sci. Fenn., 35, 473-492, (2010) · Zbl 1242.54014
[12] Illanes, A.; Nadler, S. B., Hyperspaces. fundamentals and recent advances, (1999), Marcel Dekker New York · Zbl 0933.54009
[13] Kelley, J. L., Hyperspaces of a continuum, Trans. Am. Math. Soc., 52, 1, 22-36, (1942) · Zbl 0061.40107
[14] Lynch, M., Whitney levels and certain order arc spaces, Topol. Appl., 28, 2, 189-200, (1991) · Zbl 0721.54008
[15] Martinez-Perez, A.; Morón, M. A., Uniformly continuous maps between ends of \(\mathbb{R}\)-trees, Math. Z., 263, 3, 583-606, (2009) · Zbl 1185.54029
[16] Moise, E. E., Grille decomposition and convexification theorems for compact metric locally connected continua, Bull. Am. Math. Soc., 55, 1111-1121, (1949) · Zbl 0036.11801
[17] Nadler, S. B., A characterization of locally connected continua by hyperspace retractions, Proc. Am. Math. Soc., 67, 1, 167-176, (1977) · Zbl 0376.54015
[18] Nadler, S. B., Hyperspaces of sets, Pure Appl. Math. Ser., vol. 49, (1978), Marcel Dekker, Inc. New York · Zbl 0432.54007
[19] Nadler, S. B., Continuum theory: an introduction, Pure Appl. Math. Ser., vol. 158, (1992), Marcel Dekker, Inc. New York · Zbl 0757.54009
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