On invariants and forbidden sets

Abstract

We introduce six new algebraic invariants for rational difference equations. We use these invariants to perform a reduction of order in each case. This reduction of order allows us to find forbidden sets in each case. These six cases include two linear fractional rational difference equations of order greater than one. In all six cases, we give a closed form solution for all initial conditions which are not in the forbidden set. In all six cases, the initial conditions and parameters are assumed to be arbitrary complex numbers.

On forbidden sets

“…An interesting modification of the former ideas is the use of invariants to describe the FS of some RDEs. Consider the following example from [31] x n+1 =…”

Lie symmetries of birational maps preserving genus 0 fibrations

“…This is summarized in the result below which, additionally, allows to construct the explicit conjugations. In Section 3.4.2 we use this result to detect the conjugations between the maps associated to the six recurrences presented by F. Palladino in [12].…”

“…In this section we show how the results in Section 2 can be used to analyze the global dynamics of birational maps preserving genus 0 fibrations in a unified way. The considered examples include a one-parameter family of maps previously studied by G. Bastien and M. Rogalski in [1], a map introduced by S. Saito and N. Saitoh in [13], and the maps associated to some difference equations considered by F. Palladino in [12]. The method can also be applied to study some other maps appearing in the literature, for instance the ones in [2] and [15].…”

“…Observe that for each h ≥ 2, the map M h (t) has the fixed points t j given in (12) which correspond with the infinite points Q j,h , which are fixed points of F a .…”

“…We found this difference equation when looking for a recurrence with the same invariant as the Saito and Saitoh map. Latter, we knew that this equation is a particular case of the six equations introduced in [12], and also considered in Section 3.4.…”