  We study the ergodic properties of two classes of random dynamical systems: a
type of Markov chain which we call the \textit{alternating random walk} and a
certain stochastic billiard system which describes the motion of a free-moving
rough disk bouncing between two parallel rough walls. Our main results
characterize the types of Markov transition kernels which make each system
ergodic -- in the first case, with respect to uniform measure on the state
space, and in the second case, with respect to Lambertian measure (a classic
measure from geometric optics). In addition, building on results from
\cite{rudzis2022}, we give explicit examples of rough microstructures which
produce ergodic dynamics in the second system. Both systems have the property
that the transition kernel governing the dynamics is singular with respect to
uniform measure on the state space. As a result, these systems occupy a kind of
mean in the problem space between diffusive processes, where establishing
ergodicity is relatively easy, and physically realistic deterministic systems,
where questions of ergodicity are far less approachable.
