Borel summation and splitting of separatrices for the Hénon map. (English) Zbl 0988.37031

Consider the quadratic area-preserving map of \(\mathbb{C}^2\) defined by \[ H: \binom{u}{v} \mapsto \binom {u_1=u+v-u^2} {v_1=v-u^2}. \tag{1} \] Here the origin is a parabolic fixed point of \(H\). The authors are interested in two invariant curves \(W^+\) and \(W^-\) which they call “stable” and “unstable” separatrices for the discrete-time realization of \(H\) (defined by the iteration of \(H\)). They show that the curves \(W^+\) and \(W^-\) can be naturally parametrized by a complex variable \(z\), with well-defined asymptotics as \(\operatorname {Re}z\to +\infty\). A single asymptotic series corresponds to both separatrices but in different domains of the complex plane. The intersection of these domains contains two connected components, for which the distance between the corresponding points is exponentially small as \(|\operatorname {Im} z|\to \infty\).
The main goal of the paper is to study this phenomenon asymptotically. The authors’ approach is based on an application of Borel summation to the divergent asymptotic series.


37D10 Invariant manifold theory for dynamical systems
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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