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Topology of Fatou components for endomorphisms of k : linking with the Green’s current

Suzanne Lynch Hruska, Roland K. W. Roeder (2010)

Fundamenta Mathematicae

Little is known about the global topology of the Fatou set U(f) for holomorphic endomorphisms f : k k , when k >1. Classical theory describes U(f) as the complement in k of the support of a dynamically defined closed positive (1,1) current. Given any closed positive (1,1) current S on k , we give a definition of linking number between closed loops in k s u p p S and the current S. It has the property that if lk(γ,S) ≠ 0, then γ represents a non-trivial homology element in H ( k s u p p S ) . As an application, we use these linking...

Toric structures on near-symplectic 4-manifolds

David T. Gay, Margaret Symington (2009)

Journal of the European Mathematical Society

A near-symplectic structure on a 4-manifold is a closed 2-form that is symplectic away from the 1-dimensional submanifold along which it vanishes and that satisfies a certain transversality condition along this vanishing locus. We investigate near-symplectic 4-manifolds equipped with singular Lagrangian torus fibrations which are locally induced by effective Hamiltonian torus actions. We show how such a structure is completely characterized by a singular integral affine structure on the base of...

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