  Knowledge about existence, strength, and dominant direction of causal
influences is of paramount importance for understanding complex systems. With
limited amounts of realistic data, however, current methods for investigating
causal links among different observables from dynamical systems suffer from
ambiguous results. Missing is a statistically well defined approach that avoids
false positive detections while being sensitive for weak interactions. Ideally,
it should be able to infer directed causal influences also when
synchronizations occur. The proposed method exploits local inflations of
manifolds to obtain estimates of upper bounds on the information loss among
state reconstructions from two observables. It comes with a test for the
absence of causal influences. Simulated data demonstrate that it is robust to
intrinsic noise, copes with synchronizations, and tolerates also measurement
noise.
