Pseudoholomorphic curves and the shadowing lemma
Consider a 1-periodic in time Hamiltonian system on a compact smooth manifold. Assume that the system has a hyperbolic equilibrium point \(x\) and denote by \(C\) the set of solutions doubly asymptotic to \(x\). Let \(\phi_T\) be the shift along solutions at time \(T\). The authors formulate conditions under which, for \(T\) large enough, the mapping \(\phi_T\) has a compact invariant set on which \(\phi_T\) is semiconjugate to a Bernoulli shift. Usually, properties similar to that formulated above are consequences of transversality of homoclinic trajectories. In this paper, the authors use a weaker assumption; namely, they assume that connected components of the set \(C\) are compact (with respect to a proper topology). The main result is a consequence of a special topological shadowing lemma based on the notion of a pseudoholomorphic curve.
scientific article; zbMATH DE number 1425237
scientific article