Dynamics of the third order Lyness' difference equation

This paper studies the iterates of the third order Lyness' recurrence [image omitted], with positive initial conditions, being a also a positive parameter. It is known that for a=1 all the sequences generated by this recurrence are 8-periodic. We prove that for each [image omitted] there are infinitely many initial conditions giving rise to periodic sequences which have almost all the even periods and that for a full measure set of initial conditions the sequences generated by the recurrence are dense in either one or two disjoint bounded intervals of . Finally, we show that the set of initial conditions giving rise to periodic sequences of odd period is contained in a co-dimension one algebraic variety (so it has zero measure) and that for an open set of values of a it also contains all the odd numbers, except finitely many of them.