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We prove the saturation of a generalized partially hyperbolic attractor of a C2
map. As a consequence, we show that any generalized partially hyperbolic horseshoe-like
attractor of a C1-generic dieomorphism has zero volume. In contrast, by modication of
the Poincare cross section of Lorenz geometric model, we build a C1-dieomorphism with a
partially hyperbolic horseshoe-like attractor of positive volume.
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