Semiflows induced by length metrics: On the way to extinction

TL;DR

This paper studies a semiflow on the hyperspace of a length metric Peano continuum, analyzing how it evolves from the entire space to a point over time, with applications to various classes of spaces.

Contribution

It introduces a semiflow framework on the hyperspace of length metric Peano continua, linking geometric properties with dynamical evolution.

Findings

Semiflow collapses spaces into a point over time.

Properties vary across different classes of spaces.

Framework applies to manifolds, graphs, and polyhedra.

Abstract

Bing and Moise proved, independently, that any Peano continuum admits a length metric d. We treat non-degenerate Peano continua with a length metric as evolution systems instead of stationary objects. 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.