PERIODIC-ORBITS OF DIFFERENCE-EQUATIONS

The real difference equation a(n+2)-(lambda\a(n+1)\ + mu a(n+1)) + a(n ) = 0 may be interpreted as a dynamical system Phi:(a(n), a(n+1))-->(a (n+1), a(n+2)) acting in the plane. The set Lambda(p) of points (lambd a, mu) for which the mapping Phi is periodic has a rich structure. In this paper, we derive some geometric properties of Lambda(p) (for exam ple, we show that it is unbounded and uncountable), and we derive crit eria for Phi to be periodic. We also investigate when Phi is conjugate to a rotation of the plane, and we describe how the rotation numbers of the corresponding circle maps Phi/\Phi\ are related to the structur e of Lambda(p).