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A characterization of dynamically defined zeta functions is presented. Itcomprises a list of axioms, natural extension of the one which characterizestopological degree, and a uniqueness theorem. Lefschetz zeta function is themain (and proved unique) example of such zeta functions. Another interpretationof this function arises from the notion of symmetric product from which somecorollaries and applications are obtained.

Uniqueness of dynamical zeta functions and symmetric products