Abstract
We study the QRT-maps associated with the family of biquadratic curves \(C_d(K)\) with equations \(x^2y^2 - dxy - 1 + K(x^2 + y^2) = 0\). With the Prime Number Theorem and the geometry of elliptic cubics we determine the periods of periodic orbits of the dynamical systems defined by these QRT-maps, and prove sensitivity to its initial conditions.
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Appendices
Appendix 1: 4 and 6 are Minimal Periods
(a) We will find a necessary and sufficient condition for 4 being a minimal period.
Let \({\mathcal {V}}\) and \({\mathcal {H}}\) be the involutions defined by vertical [resp. horizontal] alignment of two points on the same curve \({\mathcal {C}}_d(K)\), so that we have \(T_d={\mathcal {V}}\circ {\mathcal {H}}\) and \(T_d^{-1}={\mathcal {H}}\circ {\mathcal {V}}\) (excepted for the points \(M_0\), \(M_1\), \(N_0\), \(N_1\)).
From Proposition 11 or Corollary 12, we see that \(T_d\) is 4-periodic on \({\mathcal {C}}_d(K)\) if and only if (with an exception that we will see later) \(A_0:= (m,m)\) is 4-periodic for \(T_d\) (m is given by relation (7)), or the same condition for \(B_0:=(-m,-m)\). The condition \(T_d^4(A_0)=A_0\) can be written
But these two points are symmetric with respect to the diagonal (because the curve is symmetric). So they can be equal if and only if they are both equal to \(A_0\) or to \(B_0\). If they are equal to \(A_0\), we would have \(T_d^2(A_0)=A_0\), and \(T_d\) would be 2-periodic, which is false. So we must have the equality
If we put \(T_d(A_0)=(X,Y)\) and \(T_d^{-1}(B_0)=(X_1,Y_1)\), the condition (61) is \(X-X_1=0\) and \(Y-Y_1=0\), and the easy (with Maple) computation gives
As an example, for \(d=5\) and \(K=3\), \(T_5\) is 4-periodic on the curve \({\mathcal {C}}_5(3)\).
An exception happens when the two denominators (which are the same ones) of \(X-X_1\) and \(Y-Y_1\) are 0, that is when
But it is easy to see that relations (64) and (62) are incompatible. So this exception is not a curve on which \(T_d\) is 4-periodic.
It is also interesting to note that the condition that \({\mathcal {H}}(A_0)=N_0\), so that \(Y=\infty \), id est \(T_d(A_0)=V\), is exactly the relation (61), and by symmetry it is also the relation for having \(T_d^{-1}(B_0)=H\). So we find again the previous exception.
(b) Now we will find a condition for 6 being a minimal period.
With the same reasoning we find that the condition is
so that, by symmetry, these to points must be equal to \(A_0\) or to \(B_0\). If it would \(A_0\), \(T_d\) would be 3-periodic, and 6 would not be minimal. So the good condition is
It is now easy, with the use of Maple, and by computing \(T_d^2(A_0)=(X_2,Y_2)\), to find the condition for 6 being a minimal period (at least, it is a sufficient condition)
which gives easily d as a function of K
As an example, for \(d=4\sqrt{3+2\sqrt{3}}\) and \(K=2+\sqrt{3}\), we obtain a curve on which 6 is a minimal period.
Appendix 2: On the Forms of the Curves \({\mathcal {C}}_d(K)\) (Proof of Lemma 5)
The plan of proof of Lemma 5 is simple :
(1) The curves are starlike with respect the point (0, 0) : we put \(x=\rho u\) and \(y=\rho v\), with a unit vector (u, v). If \(uv\not =0\) we have with equation of \({\mathcal {C}}_d(K)\) a quadratic equation in \(\rho ^2\) with a unique positive solution, so \(\rho =\pm \alpha \): the curve is starlike. If \(uv = 0\), \(\rho =\pm \frac{1}{\sqrt{K}}\) if \(K>0\), and if \(K\le 0\) and \(uv=0\), there is no solution: the curves \({\mathcal {C}}_d(K)\) does not cut the axes.
(2) There is no double point in finite distance (see Lemma 6).
(3) If \(K>0\), one see the inclusion \({\mathcal {C}}_d(K)\subset B\Big ((0,0),\sqrt{\frac{1+d^2/4}{K}}\Big )\): one has \(\displaystyle x^2+y^2=\frac{1+dxy-x^2y^2}{K}\le \frac{1+d^2/4}{K}\). So, from the three previous points, one see that the curves are homeomorphic to circles for \(K>0\).
(4) If \(K<0\), it is easy to find the asymptotes of the curve \({\mathcal {C}}_d(K)\). \(\square \)
Appendix 3: Previous Particular Results of the Authors About QRT-Maps
The invention of QRT-maps was originally in [18], for physical reasons, and the essential exemple was on [22], but these papers were not easy to find. So the first papers ([3, 11] and [4]) were slightly different. They studied the now called “symmetric special QRT-maps”, where the family of biquadratic curves \({\mathcal {C}}(K)\) with equations \(Q_1(x,y)-KQ_2(x,y)=0\) were symmetric with respect the diagonal, and with \(Q_2(x,y)=xy\); and the QRT-map was defined by the following way: if \({\mathcal {C}}(K)\) is the curve passing through the point M, we cut it in \(M_1\) by the horizontal line which contains M, and then T(M) is the symmetric of \(M_1\) with respect the diagonal (it is on \({\mathcal {C}}(K)\)).
These cases correspond to the study of difference equations of the form
In [8] we studied the case of difference equations \(u_{n+2}+u_n=\psi (u_{n+1})\) which is associated with “symmetric QRT-maps” defined by families of the forms \(Q_1(x,y)+K=0\).
In each of these papers, we determine the possible periods of periodic orbits, prove the density of periodic points and not periodic points, and a form of sensitivity to initial conditions.
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 :
In [2] we studied the 2-periodic Lyness’ equation \(u_{n+2}u_n=u_{n+1}+a_n\), with \(a_n\) 2-periodic.
In [12] we study the particular case of the dynamical systems \((x,y)\mapsto T_d(x,y)= (X,Y)\) given by
with curves \({\mathcal {C}}_d(K)\) equations of them are
In the present paper we begin study of non-special QRT-dynamical systems, associated with QRT-families of curves with equations \(Q_1(x,y)-KQ_2(x,y)=0\) with \(Q_2(x,y)\) not of the form xy.
In fine, in [9] and [10] we present examples of QRT-families of degree four, but such that each of the curves of the family has genus zero, and studied the correspondent dynamical systems.
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Bastien, G., Rogalski, M. The Periodic Orbits of a Dynamical System Associated with a Family of QRT-Maps. Qual. Theory Dyn. Syst. 19, 57 (2020). https://doi.org/10.1007/s12346-020-00393-2
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DOI: https://doi.org/10.1007/s12346-020-00393-2