For volume-preserving partially hyperbolic Anosov diffeomorphisms on T^3, smooth rigidity is equivalent to the stable and center distributions exceeding their critical Hölder exponents; otherwise these distributions have fractal graphs.
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Invariant distributions of partially hyperbolic systems: fractal graphs, excessive regularity, and rigidity
For volume-preserving partially hyperbolic Anosov diffeomorphisms on T^3, smooth rigidity is equivalent to the stable and center distributions exceeding their critical Hölder exponents; otherwise these distributions have fractal graphs.