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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.

37B45 Continua theory in dynamics
54E35 Metric spaces, metrizability
54B20 Hyperspaces in general topology
Full Text: DOI arXiv
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