Problem 1744

Throughout the solution we will use the notation $x_{n+1}=x_{1}$. We prove that the set of possible values of $a$ is $$\left(-\infty, \frac{1}{4}\right] \bigcup\left\{\frac{1}{4 \cos ^{2} \frac{k \pi}{n}} ; k \in \mathbb{N}, 1 \leq k<\frac{n}{2}\right\}$$ In the case $a \leq \frac{1}{4}$ we can choose $x_{1}$ such that $x_{1}\left(1-x_{1}\right)=a$ and set $x_{1}=x_{2}=\ldots=x_{n}$. Hence we will now suppose that $a>\frac{1}{4}$. The system gives the recurrence formula $$x_{i+1}=\varphi\left(x_{i}\right)=1-\frac{a}{x_{i}}=\frac{x_{i}-a}{x_{i}}, \quad i=1, \ldots, n$$ The fractional linear transform $\varphi$ can be interpreted as a projective transform of the real projective line $\mathbb{R} \cup\{\infty\}$; the map $\varphi$ is an element of the group $\operatorname{PGL}_{2}(\mathbb{R})$, represented by the linear transform $M=\left(\begin{array}{cc}1 & -a \\ 1 & 0\end{array}\right)$. (Note that $\operatorname{det} M \neq 0$ since $a \neq 0$.) The transform $\varphi^{n}$ can be represented by $M^{n}$. A point (written in homogenous coordinates) is a fixed point of this transform if and only if $(u, v)^{T}$ is an eigenvector of $M^{n}$. Since the entries of $M^{n}$ and the coordinates $u, v$ are real, the corresponding eigenvalue is real, too. The characteristic polynomial of $M$ is $x^{2}-x+a$, which has no real root for $a>\frac{1}{4}$. So $M$ has two conjugate complex eigenvalues $\lambda_{1.2}=\frac{1}{2}(1 \pm \sqrt{4 a-1} i)$. The eigenvalues of $M^{n}$ are $\lambda_{1,2}^{n}$, they are real if and only if $\arg \lambda_{1,2}= \pm \frac{k \pi}{n}$ with some integer $k$; this is equivalent with \pm \sqrt{4 a-1}=\tan \frac{k \pi}{n}$$$$a=\frac{1}{4}\left(1+\tan ^{2} \frac{k \pi}{n}\right)=\frac{1}{4 \cos ^{2} \frac{k \pi}{n}}$$$$ If $\arg \lambda_{1}=\frac{k \pi}{n}$ then $\lambda_{1}^{n}=\lambda_{2}^{n}$, so the eigenvalues of $M^{n}$ are equal. The eigenvalues of $M$ are distinct, so $M$ and $M^{n}$ have two linearly independent eigenvectors. Hence, $M^{n}$ is a multiple of the identity. This means that the projective transform $\varphi^{n}$ is the identity; starting from an arbitrary point $x_{1} \in \mathbb{R} \cup\{\infty\}$, the cycle $x_{1}, x_{2}, \ldots, x_{n}$ closes at $x_{n+1}=x_{1}$. There are only finitely many cycles $x_{1}, x_{2}, \ldots, x_{n}$ containing the point $\infty$; all other cycles are solutions for the system.