Pseudoholomorphic curves and the shadowing lemma
type: Item
instance of: scholarly article
title: Pseudoholomorphic curves and the shadowing lemma
zbMATH Open document ID: 0955.37039
DOI: 10.1215/S0012-7094-99-09902-7
author: K. Cieliebak
author: Éric Séré
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 37C50
Mathematics Subject Classification ID: 32Q65
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 37C29
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 1425237
zbMATH Keywords: Bernoulli shift
zbMATH Keywords: homoclinic trajectories
zbMATH Keywords: shadowing lemma
zbMATH Keywords: pseudo-holomorphic curve
reviewed by: Sergei Yu. Pilyugin
MaRDI profile type: Publication
Wikidata QID: Q124883523
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full work available at URL: s0012-7094-99-09902-7
OpenAlex ID: W2051159392
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description: scientific article; zbMATH DE number 1425237