Invariants and related Liapunov functions for difference equations (Q1586305)

The author considers the relationship between continuous invariants of some nonlinear difference equations and the corresponding Lyapunov functions. The main result is the following Theorem. Consider the difference equation \[ x_{n+1} =f(x_n),\;n\in N:=\{n_0,n_0+ 1,n_0+2, \dots\}, \] where \(n_0\) is a fixed integer, \(x_n\in\mathbb{R}^k\), \(f:D\to D\) \((D\subset \mathbb{R}^k)\) is continuous. Suppose that \(I:\mathbb{R}^k \to\mathbb{R}\) is a continuous invariant that is \(I(f(x))= I(x)\), \(x\in D\). If \(I\) attains an isolated local minimum or maximum value at the equilibrium point \(p\) of this system, then \(\pm (I(x)-I(p))\) is a Lyapunov function ensuring the stability of the equilibrium point \(p\). The above theorem is then applied successively to some known important difference equations such as the Lyness' equation. Todd's equation, some generalized Lyness' equations and the Gumovski-Mira equation.