Abstract: A topologically expansive system X (one where the diagonal is repelling in X^2) has an expansive metric, where sufficiently close points are moved apart by a definite ratio. We define the expansivity to be the supremal ratio one can achieve by varying the metric. It is bounded above by the topological entropy. Finer upper bounds come from the topological entropy of invariant subgraphs, while lower bounds come from constructing metrics, which can also be made precise using techniques from graphs. In this way we find the expansivity exactly for many polynomials and rational maps. Closer investigation reveals a finer structure of degrees of expansion of the dynamical system, all bounded by the topological entropy. 

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