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We study the rich behavior of ergodicity and conservativity of Cartesian products of infinite measure-preserving transformations. A class of transformations is constructed such that for any subset R ⊂ ℚ ∩ (0, 1) there exists T in this class such that T  p  × T  q  is ergodic if and only if \(\frac{p}{q}\) ∈ R. This contrasts with the finite measure-preserving case where T  p  × T  q  is ergodic for all nonzero p and q if and only if T × T is ergodic. We also show that our class is rich in the behavior of conservative products.

On infinite transformations with maximal control of ergodic two-fold product powers