In this paper expansive homeomorphisms (continuous maps) of a compact metric space \((X,d)\) are studied. If \(f:X\rightarrow X\) is a homeomorphism (continuous map), then it is called an expansive homeomorphism (continuous map) if “there is a positive \(\gamma\) such that \(d(f^{n}(x),f^{n}(y))<\gamma\) for all integer (natural number) \(n\) implies \(x=y\)”. The supremum of such \(\gamma\) is called the expansive constant of \(f\), and denoted by \(\gamma(f)\). The author has defined \(h^{+}_{E}(f)\) and \(h^{-}_{E}(f)\) by \(\limsup_{n\rightarrow \infty} -\frac{\log\gamma(f^{n})}{n}\) and \(\liminf_{n\rightarrow \infty} -\frac{\log\gamma(f^{n})}{n}\) respectively. Interesting results on the properties of \(h^{+}_{E}(f)\) and \(h^{-}_{E}(f)\), and their relations with entropy and box dimensions are deduced.