The aim of this work is to show a use of persistent homology in the study of discrete dynamical systems. The notion of persistence is presented in a more general categorical setting compared to the standard definition of persistent homology. An algebraic correspondence between persistent modules over a field and finitely generated graded modules over the ring of polynomials in one variable is established. As a consequence, a simple description of persistence in terms of the persistent diagram is obtained. With the use of persistence of eigenspaces of an induced endomorphism on homology, a procedure is developed which allows us to describe the global behaviour of an unknown self-map just from a given finite sample of points and a sampled map. An algorithm to compute the persistent diagram of a tower of eigenspaces is given and its stability is proven. Lastly, it is tested on a few basic examples and the effect of different parameters (e.g. sample size, noise) on the results is explained.
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