Some properties of the k-dimensional Lyness' map

TL;DR

This paper investigates the properties of the k-dimensional Lyness' map, revealing symmetries, invariants, and dynamical behaviors, especially for dimensions up to five and odd dimensions, enhancing understanding of its invariant sets and periodic points.

Contribution

It introduces a Lie symmetry for the map, identifies a new first integral for odd dimensions, and analyzes the nature of invariant sets and periodic points.

Findings

A rational vector field provides a Lie symmetry for the map.

A new first integral for F^2 in odd dimensions is established.

Most positive initial conditions are not periodic points of odd period.

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.