In this paper we extend to a very general context Newhouse's phenomenon concerning the persistence of homoclinic tangencies and the coexistence of infinitely many sinks. This is done using the corresponding results in codimension one recently provedby J. Palis and M. Viana, and in a reduction of codimension in the unfolding of homoclinic tangencies developed in the present paper.

Persistence of homoclinic tangencies for area-preserving maps Annales de la faculté des sciences de Toulouse 6 e série, tome 6, n o 4 (1997), p. 711-725 © Université Paul Sabatier, 1997, tous droits réservés. L'accès aux archives de la revue « Annales de la faculté des sciences de Toulouse » (http://picard.ups-tlse.fr/~annales/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Define the quadratic family of order two as F µ (x, y) = (y, −x 2 + µx), where µ is a real parameter. The boundary of the basin of attraction of the fixed point at ∞ is an invariant curve for µ < 4, and is a Cantor set for µ > 4. Perturbations of F µ with µ = 4 were studied in Romero et al (2001 Discrete Continuous Dynam. Syst. 7 35) (also in higher dimension), where it was proved that these situations persist. Now we study perturbations of the bifurcation point µ = 4, where the explosion of the basin, B ∞ , occurs. We prove that either there exists a connected invariant curve J contained in the boundary of the basin, or the set of critical points is a subset of B ∞ and the boundary has uncountably many components accumulated by the pre-images of the analytic continuation of the fixed point at the origin. The curve J undergoes a fractalization process until it ceases to exist.

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In this article we establish the following result: if a nondegenerate quadratic endomorphism of the plane has no fixed points, then every point has empty omega-limit set and alpha-limit set. It is also shown that there exists a six parameter family open and dense in the space of all quadratic mappings of the plane (even those having fixed points). The degenerate case (when the quadratic forms of both components are linearly dependent), for which the theorem fails, is considered in the last section.

A vertical delay endomorphism F on R k , with k ≥ 2, is the endomorphism associated to the difference equationwhere the function f is C 2 and its partial derivative of second order with respect to the first variable is bigger than every other partial derivative of second order. The main goal of this paper is to describe the dynamical behaviour of a huge class F of one-parameter families of vertical delay endomorphisms. We will prove that for any {F µ}µ∈R in F and every |µ| large enough, the nonwandering set Ω(Fµ) of Fµ, is either the empty set or an expanding Cantor set and the restriction of Fµ to Ω(Fµ) is conjugated to the unilateral shift on two symbols.