Lyness' Difference Equation 
in 
, with a Particular Study of the Case 
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.