Some properties of the k-dimensional Lyness' map

Abstract

This paper is devoted to study some properties of the k-dimensional Lyness' map. Our main result presentes a rational vector field that gives a Lie symmetry for F. This vector field is used, for k less or equal to 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^2 which allows 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.

22 pages; 3 figures

References in corpus (5)

• Yang-Baxter maps and symmetries of integrable equations on quad-graphs
• Yang-Baxter maps: dynamical point of view
• Studying discrete dynamical systems trough differential equations
• Dynamics of the third order Lyness' difference equation
• Third-order integrable difference equations generated by a pair of second-order equations

Cited by in corpus (5)

• Global structure of quaternion polynomial differential equations
• Integrability and non-integrability of periodic non-autonomous Lyness recurrences
• Linear Fractional Recurrences: Periodicities and Integrability
• Meromorphic first integrals of analytic diffeomorphisms
• On the accumulation points of non-periodic orbits of a difference equation of fourth order