Invariant manifolds for delay endomorphisms
In this paper the family of delay endomorphisms \[ F_\mu(x_1,\ldots,x_k)=(x_2,\ldots,x_k,-x_1^2+\mu x_1) \] is studied. It is proved that there is a \(C^2\) neighbourhood \(\mathcal{U}\) of the family \(\{F_\mu:\mu>1\}\) in the space of \(C^2\) endomorphisms of \(\mathbb{R}^k\) such that: (a) For every \(G\in \mathcal{U}\) the point at infinity is attracting and there is a fixed point \(p\) in the boundary of its basin of attraction \(B_\infty\). The differential of \(G\) at \(p\) has a unique positive eigenvalue \(\lambda>1\). (b) If the set of critical points of \(G\in \mathcal{U}\) is included in \(B_\infty\) then \(B_\infty\) is connected and its complementary set \(A\) has uncountably many components. (c) If \(\mu<4\) and \(G\) is a small perturbation of \(F_\mu\) then the boundary of the immediate basin of attraction of \(\infty\) is a codimension one submanifold \(J_G\). Moreover, if (with the notation in (a)) \(\lambda>|\lambda_0|\) for any other eigenvalue \(\lambda_0\) of the differential of \(G\) at \(p\) then \(J_G\) is of class \(C^1\). We are referring here to the Whitney topology, that is, a \(C^2\) neighbourhood \(\mathcal{U}\) of a map \(F\) is determined for some continuous maps \(\varepsilon_i:\mathbb{R}^k\rightarrow \mathbb{R}^+\), and consists of the maps \(G\) such that \(\|D_x^2G-D_x^2F\|\leq \varepsilon_2(x)\), \(\|D_xG-D_xF\|\leq \varepsilon_1(x)\) and \(\|G(x)-F(x)\|<\varepsilon_0(x)\) for any \(x\in \mathbb{R}^k\). It is also proved that if \(G\) satisfies the hypothesis in (c) then there is an open set \(J'\subset J_G\) containing \(p\) in its boundary such that the following properties hold: (i) \(G^k(J')\subset J'\), and if \(\text{Cl}(J')\) denotes the closure of \(J'\) then \(\bigcup_{j=0}^{k} G^{-j}(\text{Cl}(J'))\supset J\). (ii) \(G^j(J')\cap J'=\emptyset\) for every \(1\leq j\leq k-1\). (iii) \(J'\) is homeomorphic and close to a \((k-1)\)-dimensional open cube \(Q_{k-1}\) and the restriction of \(G^k\) to any compact subset of \(J'\) is \(C^1\) close to \(\pi_\mu^{(k-1)}\) in \(Q_{k-1}\), with \(\pi_\mu^{(k-1)}\) denoting the \(k-1\)-dimensional product of the quadratic map \(\pi_\mu(x)=-x^2+\mu x\).
scientific article; zbMATH DE number 1882961