Rigidity of center Lyapunov exponents and $su$-integrability

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Let f be a conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism A on \mathbb{T}^3 . We show that the stable and unstable bundles of f are jointly integrable if and only if every periodic point of f admits the same center Lyapunov exponent with A . This implies every conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism on \mathbb{T}^3 , is ergodic. This proves the Ergodic Conjecture proposed by Hertz–Hertz–Ures on \mathbb{T}^3 .