Automorphism groups of positive entropy on minimal projective varieties

We determine the geometric structure of a minimal projective threefold having two 'independent and commutative' automorphisms of positive topological entropy, and generalize this result to higher-dimensional smooth minimal pairs (X, G). As a consequence, we give an effective lower bound for the first dynamical degree of these automorphisms of X fitting the 'boundary case'. (C) 2010 Elsevier Inc. All rights reserved.