  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 classify ergodicity and mixing. We also develop the
computational methods for information transfer computation and apply the
framework for optimal placements of actuators and sensors for control of
non-equilibrium dynamics.
