Clifford asymptotics and the local lefschetz index

1989 ◽  
pp. 137-142
Author(s):  
John D. Lafferty ◽  
Yanlin Yu ◽  
Weiping Zhang
Keyword(s):  
Lefschetz Index

scholarly journals On the iterated Hamiltonian Floer homology

2020 ◽  
pp. 2050026
Author(s):  
Erman Çı̇nelı̇ ◽  
Viktor L. Ginzburg
Keyword(s):  
Fixed Point ◽  
Euler Characteristic ◽  
Group Action ◽  
Floer Homology ◽  
Local Homology ◽  
Lefschetz Index ◽  
Cyclic Group Action ◽  
Symplectically Aspherical Manifold ◽  
Aspherical Manifold ◽  
Symplectically Aspherical

The focus of the paper is the behavior under iterations of the filtered and local Floer homology of a Hamiltonian on a symplectically aspherical manifold. The Floer homology of an iterated Hamiltonian comes with a natural cyclic group action. In the filtered case, we show that the supertrace of a generator of this action is equal to the Euler characteristic of the homology of the un-iterated Hamiltonian. For the local homology the supertrace is the Lefschetz index of the fixed point. We also prove an analog of the classical Smith inequality for the iterated local homology and the equivariant versions of these results.

scholarly journals Existence of periodic points near an isolated fixed point with Lefschetz index one and zero rotation for area preserving surface homeomorphisms

2015 ◽  
Vol 36 (7) ◽  
pp. 2293-2333
Author(s):  
JINGZHI YAN
Keyword(s):  
Fixed Point ◽  
Periodic Orbits ◽  
Periodic Points ◽  
Dirac Measure ◽  
Lefschetz Index ◽  
Weak Star ◽  
Surface Homeomorphisms ◽  
Rotation Set ◽  
Area Preserving ◽  
Measure Sense

Let $f$ be an orientation and area preserving diffeomorphism of an oriented surface $M$ with an isolated degenerate fixed point $z_{0}$ with Lefschetz index one. Le Roux conjectured that $z_{0}$ is accumulated by periodic orbits. In this paper, we will approach Le Roux’s conjecture by proving that if $f$ is isotopic to the identity by an isotopy fixing $z_{0}$ and if the area of $M$ is finite, then $z_{0}$ is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at $z_{0}$ is the limit in the weak-star topology of a sequence of invariant probability measures supported on periodic orbits. Our proof is purely topological. It works for homeomorphisms and is related to the notion of local rotation set.

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