  Let $\IM =\otimes_{n \in \IZ}\!M^{(n)}(\IC)$ be the two sided infinite tensor
product $C^*$-algebra of $d$ dimensional matrices $\!M^{(n)}(\IC)=\!M_d(\IC)$
over the field of complex numbers $\IC$. Let $\omega$ be a translation
invariant state of $\IM$. In a recent paper, we have proved that the mean
entropy $s(\omega)$ is a complete invariant for certain classes of translation
invariant state $\omega$ of $\IM$. In this paper, we have developed a general
theory for dynamical entropy for an automorphism on an arbitrary $C^*$- or
von-Neumann algebras based on repeated admissible measurement processes. In
particular, we prove that dynamical entropy $h_{\omega}(\theta)$ for
translation dynamics $(\IM,\theta,\omega)$ satisfies $s(\omega) \le
h_{\omega}(\theta) \le 2s(\omega)$. In case $\omega$ is an infinite tensor
product state of $\IM$ then $h_{\omega}(\theta)=s(\omega)$.
