Branch Structure Of J-Holomorphic Curves Near Periodic Orbits Of A Contact Manifold

Let M be a three–dimensional contact manifold, and ψ : D \{0} → M x ℝ a finite–energy pseudoholomorphic map from the punctured disc in ℂ that is asymptotic to a periodic orbit of the contact form. This article examines conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit such that ψ resembles a holomorphic curve, invoking comparison with the theory of topological linking of plane complex algebroid curves near a singular point. Examples of this behaviour, which are studied in some detail, include pseudoholomorphic maps into Ep,q x R, where Ep,q denotes a rational ellipsoid (contact structure induced by the standard complex structure on ℂ²), as well as contact structures arising from non-standard circle–fibrations of the three–sphere.