Combinatorial approximation by Devaney-chaotic or periodic volume preserving homeomorphisms

We modify a combinatorial idea of Peter Lax, to uniformly approximate any v olume preserving homeomorphism of the closed unit cube I-n, n greater than or equal to 2, by another one which either: (i) rigidly permutes an infinit e family of dyadic cubes of total volume one, or (ii) is chaotic in the sen se of Devaney. Our methods apply as well to arbitrary compact manifolds end owed with a nonatomic, locally positive Borel measure.