  The first-return map, or the Poincar\'e map, is a fundamental concept in the
theory of flows. However, it can generally be defined only partially, and
additional conditions are required to define it globally. Since this partiality
reflects the dynamics, the flow can be described by considering the domain and
behavior of such maps. In this study, we define the concept of first-exit maps
and first-return maps, which are partial maps derived from flows, to enable
such analysis. Moreover, we generalize some notions related to the first-return
maps. It is shown that the boundary points of an open set can be classified
based on the behavior of these maps, and that this classification is invariant
under topological equivalence. Further, we show that some dynamical properties
of a flow can be described in terms of the types of boundary points. In
particular, if the flow is planar and the open set has a Jordan curve as its
boundary, a more detailed analysis is possible, and we present results on the
conditions which restrict possible forms of the first-exit maps . Finally, as
an application of the results obtained, we consider the relationship between
flows and a class of hybrid systems.
