  We consider the problem of optimal zero-delay coding and estimation of a
stochastic dynamical system over a noisy communication channel under three
estimation criteria concerned with the low-distortion regime. The criteria
considered are (i) a strong and (ii) a weak form of almost sure stability of
the estimation error as well as (ii) quadratic stability in expectation. For
all three objectives, we derive lower bounds on the smallest channel capacity
$C_0$ above which the objective can be achieved with an arbitrarily small
error. We first obtain bounds through a dynamical systems approach by
constructing an infinite-dimensional dynamical system and relating the capacity
with the topological and the metric entropy of this dynamical system. We also
consider information-theoretic and probability-theoretic approaches to address
the different criteria. Finally, we prove that a memoryless noisy channel in
general constitutes no obstruction to asymptotic almost sure state estimation
with arbitrarily small errors, when there is no noise in the system. The
results provide new solution methods for the criteria introduced (e.g.,
standard information-theoretic bounds cannot be applied for some of the
criteria) and establish further connections between dynamical systems,
networked control, and information theory, and especially in the context of
nonlinear stochastic systems.
