  In this work we introduce a topological method for the search of fixed points
and periodic points for continuous maps defined on generalized rectangles in
finite dimensional Euclidean spaces. We name our technique "Stretching Along
the Paths" method, since we deal with maps that expand the arcs along one
direction. Our technique is also significant from a dynamical point of view, as
it allows to detect complex dynamics. In particular, we are able to prove
semi-conjugacy to the Bernoulli shift and thus positivity of the topological
entropy, the presence of topological transitivity and sensitivity with respect
to initial conditions, density of periodic points. Moreover, our approach,
although mathematically rigorous, avoids the use of sophisticated topological
theories and it is relatively easy to apply to specific models arising in the
applications. For example we have here employed the Stretching along the paths
method to study discrete and continuous-time models arising from economics and
biology.
