  Motivated by the ubiquitous sampled-data setup in applied control, we examine
the stability of a class of difference equations that arises by sampling a
right- or left-invariant flow on a matrix Lie group. The map defining such a
difference equation has three key properties that facilitate our analysis: 1)
its power series expansion enjoys a type of strong convergence; 2) the origin
is an equilibrium; 3) the algebraic ideals enumerated in the lower central
series of the Lie algebra are dynamically invariant. We show that certain
global stability properties are implied by stability of the Jacobian
linearization of dynamics at the origin. In particular global asymptotic
stability. If the Lie algebra is nilpotent, then the origin enjoys semiglobal
exponential stability.
