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 of order one difference equations in

The Periodic Orbits of a Dynamical System Associated with a Family of QRT-Maps

“…d (m), with the aim to send the point Q to the infinite vertical point ω of the curve. This is similar to that is made in paper [7]. We recall that the new conjugated of For this, the following lemma is useful.…”

“…Moreover, we will prove that the dynamical system has a form of sensitivity to initial conditions, as in previous papers (see for instance [7], [8]).…”

“…In Part 7 we establish theorem 2, with arguments which soon are for some of them in [7], and for some others in [3] and in [8]. The two cases K > 0 and K < 0 requires different types of arguments.…”

“…Now it is known that the dynamical system (K, α) → (K, α + θ d (K)) has uniform sensitivity to initial conditions with a constant δ if δ ∈]0, 1 2 [ (see [7], Proposition 25). So it is clear that Φ is uniformly continuous, and so the dynamical system (C(K 1 , K 2 ), T d ) has η-sensitivity to initial conditions for some constant η > 0, function of K 1 and K 2 : it is Theorem 2(1).…”

“…In the other works, we studied classical QRT-maps, associated with a couple of difference equations of the form u n+1 u n = f (v n ), v n+1 v n = g(u n+1 ). We give some examples : [7] In [2] we studied the 2-periodic Lyness' equation u n+2 u n = u n+1 + a n , with a n 2-periodic.…”