Dynamics of the Third Order Lyness’ Difference Equation

This paper studies the iterates of the third order Lyness’ recurrence xk+3 = (a + xk+1 + xk+2)/xk, 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 a 6= 1 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 R. Finally we show that the set of initial conditions giving rise to periodic sequences of odd period is contained in a codimension 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.