Covariant influences for discrete dynamical systems.

We develop a rigorous theory of external influences on discrete dynamical systems, going beyond the perturbation paradigm, in that the external influence need not be a small contribution. To do so, we introduce the notion of covariant influence, which is a type of influence that does not disrupt the "arrow of time" of a discrete dynamical system. We develop a theory of covariant influences both when there is a purely deterministic evolution and when randomness is involved. Subsequently, we provide necessary and sufficient conditions for the transition between states under deterministic covariant influences and necessary conditions in the presence of stochastic covariant influences, predicting which transitions between states are forbidden. Our approach, for the first time, employs the framework of resource theories, borrowed from quantum information theory, for the study of discrete dynamical systems. The laws we articulate unify the behaviour of different types of discrete dynamical systems, and their mathematical flavour makes them rigorous and checkable.