Let ϕ \phi , ψ \psi be two ergodic n-parameter flows which preserve finite probability measures on their spaces X, Y. Let T be a nullset-preserving map: X → Y X\, \to \,Y sending each ϕ \phi -orbit homeomorphically to a ϕ \phi -orbit. Then ϕ \phi , ψ \psi are called homeomorphically orbit-equivalent. For n = 1 n\, = \,1 , there has been developed a theory of such equivalence: “Loosely Bernoulli” theory. A completely parallel theory exists for higher dimensions, except that it is necessary to impose a certain natural “growth” restriction on T, a restriction which is vacuous in the case n = 1 n\, = \,1 . In this paper we carry out this program, but only for the case of zero entropy.