Abstract: Ergodic Theory deals with the study of asymptotic behaviour of a dynamical system, which is either a group action, or a flow or a map on the state space. Dynamical systems can be broadly classified into closed and open systems. In closed systems, the orbit of a point lie in the state space for all time, whereas in open systems, the orbit of a point may escape from the state space through a hole. A classic example of this escape phenomenon is in the study of the motion of a billiard ball on the table with a hole (pocket). The first account of open dynamical systems is due to Pianigiani and Yorke in 1979, who were motivated by this example. This talk is a brief historical account of the development of the related theory and some potential questions. Ideas from symbolic dynamics and arithmetic dynamics will be presented in this context.