Periodic solutions of the Poincaré functional equation: Uniqueness

Abstract:

Alexander Šarkovskiǐ has shown that the only continuous periodic functions satisfying the duplication formula $\psi (2x)=2\psi (x)^2-1$ are of the form $\psi (x)=\cos (\omega x)$, where $\omega$ is a real constant. The aim of this paper is to generalize this result as follows: Let $I$ be a compact interval, $f:I\to I$ be a continuous surjection and $k>1$ be an integer. Let $\psi :\mathbb {R}\to I$ and $\phi :\mathbb {R}\to I$ be two solutions of the Poincaré functional equation $\psi (kx)=f(\psi (x))$ with $\psi$ cosine-like and $\phi$ non-constant, continuous and periodic. We will show that there exist a positive real number $\omega$ and an integer $m$ such that $\phi (x)=\psi \big (\omega x+m\lambda /(k-1)\big )$ ($x\in \mathbb {R}$), where $\lambda$ is the principal period of $\psi$. As a tool, we employ the simultaneous Schröder-difference functional equation \begin{align*} \sigma (kx)=\pm k\sigma (x), \qquad \sigma (x+1)=\varepsilon (x)\sigma (x)+r(x), \end{align*} where $\varepsilon (x)=\pm 1$ and $r(x)\in \mathbb {Z}$ for all $x\geq 0$, and determine its solution.