# American Institute of Mathematical Sciences

January  2001, 7(1): 35-50. doi: 10.3934/dcds.2001.7.35

## Invariant manifolds for delay endomorphisms

 1 Universidad de La República, Facultad de Ciencias. Centro de Matemática, Iguá 4225. Montevideo 11400, Uruguay 2 Universidad Politéecnica de Cataluña, Departament de Matemática Aplicada 2, Escola Técnica Superior d'Enginyers Industrials. Colom 11, 08222. Terrasa, Barcelona, Spain 3 Universidad Centro Occidental Lisandro Alvarado, Departamento de Matemática, Decanato de Ciencias. Apartado Postal 400. Barquisimeto, Venezuela

Received  October 1999 Revised  September 2000 Published  November 2000

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.
Citation: Rovella Alvaro, Vilamajó Francesc, Romero Neptalí. Invariant manifolds for delay endomorphisms. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 35-50. doi: 10.3934/dcds.2001.7.35
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