We define and study new invariants called pre-image entropies which are similar to the standard notions of topological and measuretheoretic entropies. These new invariants are only non-zero for noninvertible maps, and they give a quantitative measurement of how far a given map is from being invertible. We obtain analogs of many known results for topological and measure-theoretic entropies. In particular, we obtain product rules, power rules, analogs of the Shannon-Breiman-McMillan Theorem, the

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... rgodic decomposition of entropy, and a variational principle.