Discrete and continuous topological dynamics: Fields of cross sections and expansive flows

In this article we consider the general problem of translatingdefinitions and results fromthe category of discrete-time dynamical systems to the category of flows.We consider the dynamics of homeomorphisms and flows on compact metric spaces,in particular Peano continua.As a translating tool, we construct continuous, symmetric and monotonousfields of local cross sections for an arbitrary flow without singular points.Next, we use this structure in the study of expansive flows on Peano continua.We show that expansive flows have not stable points andthat every point contains a non-trivial continuum in its stable set.As a corollary we obtain that noPeano continuum with an open set homeomorphic to the planeadmits an expansive flow.In particular, compact surfaces do not admit expansive flows without singular points.