  This paper concerns the link between the dynamical behaviour of a dynamical
system and the dynamical behaviour of its numerical simulations. Here, we model
numerical truncation as a spatial discretization of the system. Some previous
works on well chosen examples show that the dynamical behaviours of dynamical
systems and of their discretizations can be quite different. In this paper we
are interested in generic homeomorphisms of compact manifolds. So our aim is to
tackle the following question: can the dynamical properties of a generic
homeomorphism be detected on the spatial discretizations of this homeomorphism?
We study both conservative and dissipative homeomorphisms (i.e. with or without
the assumption that the homeomorphism preserves a given measure). We will prove
that the dynamics of a single discretization of a generic conservative
homeomorphism does not depend on the homeomorphism itself, but rather on the
grid used for the discretization. Therefore, dynamical properties of a given
generic conservative homeomorphism cannot be detected using a single
discretization. Nevertheless, we will also prove that some dynamical features
of a generic conservative homeomorphism (such as the set of the periods of all
periodic points) can be read on a sequence of finer and finer discretizations.
The dissipative case is somehow opposite: we will see that the dynamics of a
generic dissipative homeomorphism is well-approximated by the dynamics of any
fine enough discretization of this homeomorphism.
