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In this article we consider the general problem of translating definitionsand results from the category of discrete-time dynamical systems to thecategory of flows. We consider the dynamics of homeomorphisms and flows oncompact metric spaces, in particular Peano continua. As a translating tool, weconstruct continuous, symmetric and monotonous fields of local cross sectionsfor an arbitrary flow without singular points. Next, we use this structure inthe study of expansive flows on Peano continua. We show that expansive flowsadmit no stable point and that every point contains a non-trivial continuum inits stable set. As a corollary we obtain that no Peano continuum with an openset homeomorphic with the plane admits an expansive flow. In particular compactsurface admits no expansive flow without singular points.

Discrete and Continuous Topological Dynamics: Fields of Cross Sections and Expansive Flows