On Information Transfer in Dynamical Systems with applications in Control of Non-equilibrium Dynamics

In this paper, we develop the concept of information transfer between the Borel-measurable sets for a dynamical system described by a measurable space and a non-singular transformation. The concept is based on how Shannon entropy is transferred between the measurable sets, as the dynamical system evolves. We show that the proposed definition of information transfer satisfies the usual notions of information transfer and causality, namely, zero transfer and transfer asymmetry. Furthermore, we show how the information transfer measure can be used to characterize ergodicity and mixing in dynamical systems. We also develop the computational methods for information transfer computation and apply the framework for optimal placement of actuators and sensors for control of non-equilibrium dynamics.