Dynamics of the third order Lyness' difference equation

Abstract

This paper studies the iterates of the third order Lyness' recurrence $x_{k+3}=(a+x_{k+1}+x_{k+2})/x_k,$ 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 $a\ne1$ 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.