  This paper considers the problem of stabilizing a discrete-time non-linear
stochastic system over a finite capacity noiseless channel. Our focus is on
systems which decompose into a stable and unstable component, and the stability
notion considered is asymptotic ergodicity of the $\mathbb{R}^N$-valued state
process. We establish a necessary lower bound on channel capacity for the
existence of a coding and control policy which renders the closed-loop system
stochastically stable. In the literature, it has been established that under
technical assumptions, the channel capacity must not be smaller than the
logarithm of the determinant of the system linearization, averaged over the
noise and ergodic state measures. In this paper, we establish that for systems
with a stable component, it suffices to consider only the unstable dimensions,
providing a refinement on the general channel capacity bound for a large class
of systems. The result is established using the notion of stabilization
entropy, a notion adapted from invariance entropy, used in the study of
noise-free systems under information constraints.
