Combinatorial approximation by Devaney-chaotic or periodic volume preserving homeomorphisms

We modify a combinatorial idea of Peter Lax, to uniformly approximate any volume 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 infinite family of dyadic cubes of total volume one, or (ii) is chaotic in the sense of Devaney. Our methods apply as well to arbitrary compact manifolds endowed with a nonatomic, locally positive Borel measure.