A QRT-system of two order one homographic difference equations: conjugation to rotations, periods of periodic solutions, sensitiveness to initial conditions

Abstract

We study the homographic system u(n+1)u(n)=c+d/v(n), v(n+1)v(n)=c+d/u(n+1) in the positive quadrant. The orbit of a point is contained in an invariant cubic curve, and the restriction to the positive part of this cubic of the associated dynamical system is conjugated to a rotation on the circle. For a dense invariant set of initial points the solutions are periodic, and if c=1 (this is always possible) every integer n\geq N(d) is the minimal period of some periodic solution. Every n\geq 11 is the minimal period of some solution for some d>0, and we find exactly the set of such minimal periods between 2 and 10. The associated dynamical system has sensitiveness to initial conditions on every compact set not containing the fixed point.