Entropy for smooth abelian actions

We generalize the usual notions of metric and topological entropy for flows to actions of compactly generated abelian Lie groups. Unlike previous generalizations, ours is nontrivial for smooth actions of ${{\mathbf{R}}^n}$, $n \geqslant 2$. We prove some elementary properties of our definitions and we relate them to characteristic exponents and the entropy conjecture.