  We introduce the notion of a quasistatic dynamical system, which generalizes
that of an ordinary dynamical system. Quasistatic dynamical systems are
inspired by the namesake processes in thermodynamics, which are idealized
processes where the observed system transforms (infinitesimally) slowly due to
external influence, tracing out a continuous path of thermodynamic equilibria
over an (infinitely) long time span. Time-evolution of states under a
quasistatic dynamical system is entirely deterministic, but choosing the
initial state randomly renders the process a stochastic one. In the
prototypical setting where the time-evolution is specified by strongly chaotic
maps on the circle, we obtain a description of the statistical behaviour as a
stochastic diffusion process, under surprisingly mild conditions on the initial
distribution, by solving a well-posed martingale problem. We also consider
various admissible ways of centering the process, with the curious conclusion
that the "obvious" centering suggested by the initial distribution sometimes
fails to yield the expected diffusion.
