  In this paper we provide an extension for the method of Discrete Lagrangian
Descriptors with the purpose of exploring the phase space of unbounded maps.
The key idea is to construct a working definition, that builds on the original
approach introduced in Lopesino et al. (2015), and which relies on stopping the
iteration of initial conditions when their orbits leave a certain region in the
plane. This criterion is partly inspired by the classical analysis used in
Dynamical Systems Theory to study the dynamics of maps by means of escape time
plots. We illustrate the capability of this technique to reveal the geometrical
template of stable and unstable invariant manifolds in phase space, and also
the intricate structure of chaotic sets and strange attractors, by applying it
to unveil the phase space of a well-known discrete time system, the H\'enon
map.
