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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...

Transitive flows on manifolds.

Víctor Jiménez López, Gabriel Soler López (2004)

Revista Matemática Iberoamericana

In this paper we characterize manifolds (topological or smooth, compact or not, with or without boundary) which admit flows having a dense orbit (such manifolds and flows are called transitive) thus fully answering some questions by Smith and Thomas. Name

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