  A dynamical model consists of a continuous self-map $T: \mathcal{X} \to
\mathcal{X}$ of a compact state space $\mathcal{X}$ and a continuous
observation function $f: \mathcal{X} \to \mathbb{R}$. This paper considers the
fitting of a parametrized family of dynamical models to an observed real-valued
stochastic process using empirical risk minimization. The limiting behavior of
the minimum risk parameters is studied in a general setting. We establish a
general convergence theorem for minimum risk estimators and ergodic
observations. We then study conditions under which empirical risk minimization
can effectively separate the signal from the noise in an additive observational
noise model. The key, necessary condition in the latter results is that the
family of dynamical models has limited complexity, which is quantified through
a notion of entropy for families of infinite sequences. Close connections
between entropy and limiting average mean widths for stationary processes are
established.
