A smooth zero-entropy diffeomorphism whose product with itself is loosely Bernoulli

Let M be a smooth compact connected manifold of dimension d ≥ 2, possibly with boundary, that admits a smooth effective 𝕋^2 -action S = {S_α ,β}_(α ,β) ∈𝕋^2 preserving a smooth volume v , and let B be the C ∞ closure of {h ∘S_α ,β∘h^- 1:h ∈Diff^∞(M,v),(α ,β) ∈𝕋^2}. We construct a weakly mixing C ∞ diffeomorphism T ∈ B with topological entropy 0 such that T × T is loosely Bernoulli. Moreover, we show that the set of such T ∈ B contains a dense G δ subset of B . The proofs are based on a two-dimensional version of the approximation-by-conjugation method.