We study the following problem: Given a stochastic nonlinear system controlled over a possibly noisy communication channel, what is the largest class of such channels for which there exist coding and control policies so that the closed-loop system is stochastically stable? The stability criterion considered is asymptotic mean stationarity (AMS). We first develop a general method (based on ergodic theory) to derive fundamental lower bounds on the channel capacity necessary for achieving asymptotic mean stationarity. These bounds are consistent, and more refined in comparison, with the bounds obtained earlier via information-theoretic methods. Moreover, our approach is more versatile in view of the models considered and allows for finer lower bounds when the AMS measure is known to admit further properties such as moment constraints.