  Discovery of causal relations is fundamental for understanding the dynamics
of complex systems. While causal interactions are well defined for acyclic
systems that can be separated into causally effective subsystems, a
mathematical definition of gradual causal interaction is still lacking for
nonseparable dynamical systems. The solution proposed here is analytically
tractable for time discrete chaotic maps and is shown to fulfill basic
requirements for causality measures. It implies a method for determination of
directed effective influences using pairs of measurements from dynamical
systems. Applications to time series from systems of coupled differential
equations and linear stochastic systems demonstrate its general utility.
