This paper is devoted to study some properties of the k-dimensional Lyness' map F(x1, ..., xk) = (x2, ..., xk, (a + ∑ki=2xi)/x1). Our main result presents a rational vector field that gives a Lie symmetry for F. This vector field is used, for k ≤ 5, to give information about the nature of the invariant sets under F. When k is odd, we also present a new (as far as we know) first integral for F○F which allows us to deduce in a very simple way several properties of the dynamical system generated by F. In particular for this case we prove that, except on a given codimension one algebraic set, none of the positive initial conditions can be a periodic point of odd period. © 2008 IOP Publishing Ltd.
|Journal||Journal of Physics A: Mathematical and Theoretical|
|Publication status||Published - 18 Jul 2008|