Entropy, products, and bounded orbit equivalence

Abstract We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) non-locally-finite product infinite groups, then the have same topological entropy. This fact used to show uniquely ergodic probability-measure-preserving such boundedly equivalent measure Our arguments based relativization property SC approximations yield more general inequalities.