Vorlesungen über Topologie

Abstract. We use a notion (introduced in Topology 41 (2002), 1119-1212), which is stronger than the concept of filtration pair, to prove a stable/unstable manifold general theorem for local homeomorphisms of the plane in a neighborhood of an isolated fixed point. Introduction and preliminary definitionsThe stable and unstable manifold theorem for hyperbolic diffeomorphisms plays a very important role in differential dynamics. At the topological level Baldwin and Slaminka proved, in [1], a stable/unstable manifold theorem for area-and orientationpreserving homeomorphisms of orientable 2-manifolds having isolated fixed points of index less than 1.There are many papers in the literature relating the fixed-point index of a homeomorphism f in a neighborhood of an isolated fixed point, and the local dynamical behavior of f . There are results in both directions, i.e. bounds (or explicit computation) for the fixed-point index from dynamical properties of f and how the knowledge of the fixed-point index can be applied to describe the dynamics locally. We will mention besides and Shub and Sullivan [22], for their relations with the present paper. Their results frequently deal with orientation-preserving homeomorphisms. The main reason for this assumption is the application of some version of Brouwer's lemma on translation arcs, see [5] or [7].The aim of this paper is to use the fixed-point index to obtain information about the dynamical behavior of a planar local homeomorphism in a neighborhood of an

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“…that contains p is a Jordan domain, by a well-known theorem of Kerékjártó (see [18]), which means that its adherence is a disk N.…”

Section: Be a Homeomorphism And Let Us Suppose That There Is A Disk Dmentioning

confidence: 99%

A stable/unstable manifold theorem for local homeomorphisms of the plane

Portal

Salazar

2005

Ergod. Th. Dynam. Sys.

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“…It has been studied in [7], [3]» [9] (see also [1], [2]). The previous axiomatic definition of it belongs to M. Jurchescu, [5].…”

Section: Rvmentioning

confidence: 99%

Cauchy Structures and the Kerékjârtô - Stoilow Compactification

Ceterchi¹

1988

Demonstratio Mathematica

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“…Since N is orientable, the classification theory of B. Kerékjártó [7] and I. Richards [12] can be stated as follows. Surfaces of finite type as defined in §1 can be broken down into connected sums of certain canonical surfaces.…”

Section: Figurementioning

confidence: 99%