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Relative asymptotic behavior of pseudoholomorphic half‐cylinders

In this article we study the asymptotic behavior of pseudoholomorphic half‐cylinders that converge exponentially to a periodic orbit of a vector field defined by a framed stable Hamiltonian structure. Such maps are of central interest in symplectic field theory and its variants (symplectic Floer homology, contact homology, and embedded contact homology). We prove a precise formula for the asymptotic behavior of the “difference” of two such maps, generalizing results from [6, 7, 12, 15]. Using this result with a technique from [14], we then show that a finite collection of pseudoholomorphic half‐cylinders asymptotic to coverings of a single periodic orbit is smoothly equivalent to solutions to a linear equation. © 2007 Wiley Periodicals, Inc.