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A Brouwer-like theorem for orientation reversing homeomorphisms of the sphere

Marc Bonino (2004)

Fundamenta Mathematicae

We provide a topological proof that each orientation reversing homeomorphism of the 2-sphere which has a point of period k ≥ 3 also has a point of period 2. Moreover if such a k-periodic point can be chosen arbitrarily close to an isolated fixed point o then the same is true for the 2-periodic point. We also strengthen this result by proving that if an orientation reversing homeomorphism h of the sphere has no 2-periodic point then the complement of the fixed point set can be covered by invariant...

A topological characterization of holomorphic parabolic germs in the plane

Frédéric Le Roux (2008)

Fundamenta Mathematicae

J.-M. Gambaudo and É. Pécou introduced the "linking property" in the study of the dynamics of germs of planar homeomorphisms in order to provide a new proof of Naishul's theorem. In this paper we prove that the negation of the Gambaudo-Pécou property characterizes the topological dynamics of holomorphic parabolic germs. As a consequence, a rotation set for germs of surface homeomorphisms around a fixed point can be defined, and it turns out to be non-trivial except for countably many conjugacy classes....

Attractors with vanishing rotation number

Rafael Ortega, Francisco Ruiz del Portal (2011)

Journal of the European Mathematical Society

Given an orientation-preserving homeomorphism of the plane, a rotation number can be associated with each locally attracting fixed point. Assuming that the homeomorphism is dissipative and the rotation number vanishes we prove the existence of a second fixed point. The main tools in the proof are Carath´eodory prime ends and fixed point index. The result is applicable to some concrete problems in the theory of periodic differential equations.

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