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On the index problem of $C^1$-generic wild homoclinic classes in dimension three

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  • We study the dynamics of homoclinic classes on three dimensional manifolds under the robust absence of dominated splittings. We prove that, $C^1$-generically, if such a homoclinic class contains a volume-expanding periodic point, then it contains a hyperbolic periodic point whose index (dimension of the unstable manifold) is equal to two.
    Mathematics Subject Classification: Primary: 37D25, 3730, Secondary: 34D09.


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