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Reparametrizations of Continuous Paths
#
Reparametrizations of Continuous Paths

##
Ulrich Fahrenberg and Martin Raussen

A reparametrization (of a continuous path) is given by a surjective weakly
increasing self-map of the unit interval. We show that the monoid of
reparametrizations (with respect to compositions) can be understood via
``stop-maps'' that allow to investigate compositions and factorizations,
and we compare it to the distributive lattice of countable subsets of the
unit interval. The results obtained are used to analyse the space of traces
in a topological space, i.e., the space of continuous paths up to
reparametrization equivalence. This space is shown to be homeomorphic to
the space of regular paths (without stops) up to increasing
reparametrizations. Directed versions of the results are important in
directed homotopy theory.

Journal of Homotopy and Related Structures, Vol. 2(2007), No. 2, pp. 93-117