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Suppose we are given a sample of a discrete dynamical system φ: M → M. Can we infer φ from this sample? Classical persistent homology can be used to analyze the sample and infer the homology of M. In this talk, we employ the machinery of Persistent Local Systems (Patel and MacPherson) to not only recover the persistent homology of the sample but also extract persistent information about the dynamics, all summarized in what we call the Generalized Persistence Diagram (Patel). Moreover, generalized persistence diagrams satisfy Bottleneck Stability, just like classical persistence. This implies that, under suitable assumptions on M and φ, the persistent dynamical information is stable with respect to perturbations of the sample.