Abstract
We study order Lyness' difference equation in
, with
and the associated dynamical system
in
. We study its solutions (divergence, permanency, local stability of the equilibrium). We prove some results, about the first three invariant functions and the topological nature of the corresponding invariant sets, about the differential at the equilibrium, about the role of 2-periodic points when
is odd, about the nonexistence of some minimal periods, and so forth
and discuss some problems, related to the search of common period to all solutions, or to the second and third invariants. We look at the case
with new methods using new invariants for the map
and state some conjectures on the associated dynamical system in
in more general cases.