Theorem 4.8 in the book "discrete dynamical system" gives a sufficient condition for an equilibrium to be locally stable in a nonlinear difference system.
It basically says the following: a nonlinear difference system given by $x_{t+1}=\phi(x_t)$ with equilibrium $x^*$ is locally stable if
- the absolute value of all eigenvalues of the Jacobian matrix $D\phi(x^*)$ is strictly less than $1$
- $\phi$ is a $C^1$ differeomorphism on $R^n$.
The second condition here is both strong and restrictive. Intuitively, since we only care about local stability, should it suffice to just require $\phi$ be a $C^1$ differeomorphism on a neighborhood of the equilibrium? Could any one confirm this intuition, and if possible, give a reference?
Besides, does $\phi$ being locally $C^1$ differeomorphism together with absolute value of one eigenvalue of the Jacobian is strictly bigger than $1$ guarantee that the equilibrium being locally unstable? (assume the equilibrium is a hyperbolic fixed point)