Branch structure of $J$-holomorphic curves near periodic orbits of a contact manifold

Let $M$ be a three-dimensional contact manifold, and $\tilde{\psi}:D\setminus\{0\}\to M\times{\mathbb{R}}$ a finite-energy pseudoholomorphic map from the punctured disc in ${\mathbb{C}}$ 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 $\tilde{\psi}$ 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 ${\mathbb{E}}_{p,q}\times{\mathbb{R}}$, where ${\mathbb{E}}_{p,q}$ denotes a rational ellipsoid (contact structure induced by the standard complex structure on ${\mathbb{C}}^{2}$), as well as contact structures arising from non-standard circle-fibrations of the three-sphere.