The periodic orbits of a dynamical system associated with a family of QRT-maps (Q2173691)

The authors study the QRT-maps associated with the family of biquadratic curves \(C_{d} (K)\) with equations \[ x^2 y^2 - dxy - 1 + K(x^2 + y^2) = 0, \] where \(d>0\) and \(K\in \mathbb{R}\). By using the Prime Number Theorem and the geometry of elliptic cubics, they determine the periods of periodic orbits of the dynamical systems defined by these QRT-maps, and prove sensitivity to initial conditions. The main results of the paper are: Theorem 1. (1) For any \(d > 0\) and \(K\neq 0\), the map \(T_d |_{C_{d}(K)}\) is conjugated to a rotation on the unit circle. Moreover, both the levels \(K\neq 0\) for which all the initial conditions give rise to trajectories that fill densely \(C_{d} (K)\) and the levels for which this curve is filled by periodic points (with the same period), are dense. As a consequence, the periodic points for \(T_{d}\) are dense in \(\mathbb{R}^{2}\), and also the non-periodic ones. (2) For every \(d > 0\), there exists an integer \(N(d)\) such that every integer \(n \geq N(d)\) is a minimal period of some point for the QRT-map \(T_{d}\). (3) Every integer \(n \geq 3\) is the minimal period for \(T_{d}\) of some initial point for some \(d\). (4) There is no point of period \(2\), nor real finite fixed point, but it is possible to extend continuously \(T_{d}\) to \((0, 0)\), with image \((0, 0)\). Theorem 2. The dynamical system associated with the QRT-map \(T_{d}\) is sensitive to initial conditions. More precisely: (1) For every \(0 < K_1 < K_2\), there exists a constant \(k_d (K_1, K_2) > 0\) such that for every \(K \in [K_1, K_2]\), for every point \(M_{0} \in C_{d} (K)\) and every neighbourhood \(V\) of \(M_{0}\) there exists a point \(M_{1} \in V\) such that \[d\left(T^{n}_{d}(M_{1}), T^{n}_{d}(M_{0})\right)> k_{d}(K_1, K_2)\] for an infinite values of \(n\) (uniform sensitivity); (2) For \(K < 0\), for every point \(M_{0}\) of \(C_{d} (K)\) there exists a constant \(k_d (M_0)\) such that for every neighbourhood \(V\) of \(M_{0}\) there exists a point \(M_{1} \in V\) such that \[d\left(T^{n}_{d}(M_{1}), T^{n}_{d}(M_{0})\right)> k_{d}(M_{0})\] for infinitely many values of \(n\) (pointwise sensitivity).