The notion ofdynamical entropy for actions of a countable free abelian group $G$ byautomorphisms of $C^*$-algebras is studied. These results are applied toBogoliubov actions of $G$ on the CAR-algebra. It is shown that the dynamicalentropy of Bogoliubov actions is computed by a formula analogous to thatfound by Størmer and Voiculescu in the case $G={\bf Z}$, and also it isproved that the part of the action corresponding to a singular spectrum giveszerocontribution to the entropy. The case of an infinite number of generators hassome essential differences and requires new arguments.