On the Topological Nature of the Stable Sets Associated to the Second Invariant of the Order $q$ Standard Lyness’ Equation

Abstract

We prove a conjecture asserted in a previous paper (see [2]) about order $q$ Lyness difference equation in ${\mathbb R}_*^+$:
$u_{n+q}\,u_n=a+u_{n+q-1}+{\dots}+u_{n+1}$, with $a>0$. It is known that the function on ${{\mathbb R}_*^+}^q$ defined by
\begin{multline*}
H(x)=\frac{(1+x_1+x_2)(1+x_2+x_3)\dots}{x_1\dots x_q}\\
\frac{(1+x_{q-1}+x_q)(a+x_1x_q+x_1+x_2+{\dots}+x_q)}{ x_1\dots
x_q}
\end{multline*}
is an {\sl invariant} for this equation. It is conjectured in [2] (and proved for $q=3$ only) that if $M>M_a:=\min H$ is {\sl sufficiently near} to $M_a$, then the set $S(M):=\{x\,|\,H(x)=M\}$ is homeomorphic to the sphere $\mathbbS$$^{^{q-1}}$. Here we prove this conjecture for every $q\geq 3$, and deduce from it that if the equilibrium $L$ of the map $T$ associated to the Lyness' equation, where $H$ attains its minimum, is for some $q$ {\sl the only critical point} of $H$, then the sets $S(M)$ are, for this $q$, homeomorphic to $\mathbbS$$^{^{q-1}}$ for every $M>M_a$, and that this is the case for $q$=3, 4 or 5.

2000 Mathematics Subject Classification. 37E, 39A10, 58F20