A bound for the fixed point index of area-preserving homeomorphisms of two-manifolds

Pelikan, Stephan; Slaminka, Edward E.

doi:10.1017/s0143385700004132

Cited by 25 publications

(18 citation statements)

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“…If we drop the hypothesis about isolation of the fixed point, it remains true for the planar case that the index of a fixed point is less than or equal to 1 Provided that the homeomorphism is area-preserving. This result was proved by Pelikan and Slaminka in [PS87]. Previously, Nikishin and Simon had addressed the same question for diffeomorphisms, see [Ni74] and [Si74].…”

Section: Franks Introduced Conley Index Techniques Inmentioning

confidence: 72%

About the homological discrete Conley index of isolated invariant acyclic continua

2013

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This article includes an almost self-contained exposition on the discrete Conley index and its duality. We work with a locally defined homeomorphism f in R d and an acyclic continuum X, such as a cellular set or a fixed point, invariant under f and isolated. We prove that the trace of the first discrete homological Conley index of f and X is greater than or equal to -1 and describe its periodical behavior. If equality holds then the traces of the higher homological indices are 0. In the case of orientation-reversing homeomorphisms of R 3 , we obtain a characterization of the fixed point index sequence {i(f n , p)} n≥1 for a fixed point p which is isolated as an invariant set. In particular, we obtain that i(f, p) ≤ 1. As a corollary, we prove that there are no minimal orientation-reversing homeomorphisms in R 3 .

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Section: Franks Introduced Conley Index Techniques Inmentioning

confidence: 72%

About the homological discrete Conley index of isolated invariant acyclic continua

2013

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“…Le Calvez [5] showed that if the associated fixed point index is greater than one, then there is a wandering domain in any neighborhood of that point. In particular, should the homeomorphism be area-preserving, this index must be lower or equal than one, a former result by Pelikan and Slaminka [8]. In the case of stable fixed points, Dancer and Ortega [2] showed this index to be exactly one; it immediately implies that isolated minimizers are unstable because they are well known to have fixed point index minus one.…”

Section: Introductionmentioning

confidence: 87%

Invariant manifolds near a minimizer

Ureña

2007

Journal of Differential Equations

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A classical result, studied, among others, by Carathéodory [C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, Chelsea, New York, 1989], states that, for second-order, scalar equations, nondegenerate periodic minimizers are hyperbolic. Consequently, the Stable/Unstable Manifold Theorem applies, and implies that, at least locally, the stable and unstable sets are regular curves intersecting transversally at the nondegenerate minimizer.For analytic equations, there is a version of this fact which holds for isolated, but possibly degenerate, minimizers.

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“…This operation produces lots of periodic points which accumulate in the fixed one. Incidentally, notice that, in contrast, if the map is a homeomorphism and the fixed point is accumulated by Per(f ) but not by Fix(f n ) then i(f n , p) = 1 (see [PS87] and also [L03], pag. 145).…”

Section: Introductionmentioning

confidence: 99%