  A dynamical system may be defined by a simple transition law - such as a map
or a vector field. Most learning techniques primarily try to recreate the
dynamic evolution law. This is a major shortcoming, as most dynamic properties
of interest are asymptotic properties such as an attractor or invariant
measure. One of the major theoretical challenges for numerical methods is that
approximating the dynamical law may not be sufficient to approximate these
asymptotic properties. This article presents a method of representing a
discrete-time deterministic dynamical system as a Markov process. The procedure
is completely data-driven. The technique is proved to be convergent -- the
stationary density of the Markov process has a support that converges to the
targeted invariant set. Thus invariant sets of arbitrary dynamical systems,
even with complicated non-smooth topology, can be approximated by this
technique. Under further assumptions of stochastic stability of the targeted
system, the technique is also shown to provide a convergent statistical
approximation.
