  We consider two numerical entropy--type invariants for actions of $\Zk$,
invariant under a choice of generators and well-adapted for smooth actions
whose individual elements have positive entropy. We concentrate on the maximal
rank case, i.e. $\Zk,\,k\ge 2$ actions on $k+1$-dimensional manifolds. In this
case we show that for a fixed dimension (or, equivalently, rank) each of the
invariants determines the other and their values are closely related to
regulators in algebraic number fields. In particular, in contrast with the
classical case of $\mathbb Z$ actions the entropies of ergodic maximal rank
actions take only countably many values. Our main result is the dichotomy that
is best expressed under the assumption of weak mixing or, equivalently, no
periodic factors: either both invariants vanish, or their values are bounded
away from zero by universal constants. Furthermore, the lower bounds grow with
dimension: for the first invariant (the Fried average entropy) exponentially,
and for the second (the slow entropy) linearly.
