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We say that a subset of a symplectic manifold is symplectically (neighbourhood) excisable if its complement is symplectomorphic to the ambient manifold, (through a symplectomorphism that can be chosen to be the identity outside an arbitrarily small neighbourhood of the subset). We use time-independent Hamiltonian flows, and their iterations, to show that certain properly embedded subsets of noncompact symplectic manifolds are symplectically neighbourhood excisable: a ray, a Cantor brush, a box with a tail, and -- more generally -- epigraphs of lower semi-continuous functions; as well as a ray with two horns, and -- more generally -- open-rooted finite trees.
We prove that any noncompact symplectic manifold which admits a properly embedded ray with a wide neighborhood is symplectomorphic to the complement of the ray by constructing an explicit symplectomorphism in the case of the standard Euclidean space.
Let $H(q,p)$ be a Hamiltonian on $T^*T^n$. We show that the sequence $H_{k}(q,p)=H(kq,p)$ converges for the $gamma$ topology defined by the author, to $bar{H}(p)$. This is extended to the case where only some of the variables are homogenized, that is
We study the geometry of manifolds carrying symplectic pairs consisting of two closed 2-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.
We give a method to resolve 4-dimensional symplectic orbifolds making use of techniques from complex geometry and gluing of symplectic forms. We provide some examples to which the resolution method applies.
We prove a version of the Arnold conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $beta$ cannot be disjoined from itself by a $C^0