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Invariant manifolds for delay endomorphisms

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  • Let $F_\mu(x_1,\cdots,x_k)=(x_2,\cdots,x_k,-x_1^2+\mu x_1)$. For any $G$ in a $C^2$ neighborhood $\mathcal{U}$ of the family $F_\mu$, the point at $\infty$ is an attractor (with basin denoted by $B_\infty$), and there exists a repelling fixed point in the boundary of $B_\infty$. This gives the initial step to the study of the whole boundary of $B_\infty$ and the changes it suffers: for perturbations of $F_\mu$ with $\mu$ small, the boundary of $B_\infty$ is an invariant codimension one manifold, while for large values of $\mu$ the basin $B_\infty$ is dense and its complementary set an expanding Cantor set. The techniques developed will be applied to delay endomorphisms.
    Mathematics Subject Classification: 37C05, 37D10, 37D20.


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