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For a
$C^{1+\alpha }$
diffeomorphism f of a compact smooth manifold, we give a necessary and sufficient condition that guarantees that if the set of hyperbolic Lyapunov–Perron regular points has positive volume, then f preserves a smooth measure. We use recent results on symbolic coding of
$\chi $
-non-uniformly hyperbolic sets and results concerning the existence of SRB measures for them.
