The generalized Lefschetz number of homeomorphisms on punctured disks

Un indice qui affine l'indice de Poincaré-Lefschetz pour les homéomorphismes de surfaces

We study the dynamics of surface homeomorphisms around isolated fixed points whose Poincaré-Lefschetz index is not equal to 1. We construct a new conjugacy invariant, which is a cyclic word on the alphabet $\{\ua, \ra, \da, \la\}$. This invariant is a refinement of the P.-L. index. It can be seen as a canonical decomposition of the dynamics into a finite number of sectors of hyperbolic, elliptic or indifferent type. The contribution of each type of sector to the P.-L. index is respectively -1/2, $+1/2$ and 0. The construction of the invariant implies the existence of some canonical dynamical s…

Existence of periodic points near an isolated fixed point with Lefschetz index $1$ and zero rotation for area preserving surface homeomorphisms

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 article, 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 weak-star topology of a sequence of invaria…

广义勒夫谢茨数与环辫群

The generalized Lefschetz number and loop braid groups

本文通过引入环辫群和广义勒夫谢茨数，研究三维流形中同胚的固定点和周期点的拓扑动力学性质。

穿孔圆盘上的强Nielsen等价性

Strong Nielsen equivalence on the punctured disc

本文研究穿孔圆盘上保持定向同胚的周期点强Nielsen等价性，在辫子理论中给出两个周期点强Nielsen等价的充要条件，并应用于Jiang--Zheng迹公式，证明强制周期轨道分属不同强Nielsen类。

Finite orbits for nilpotent actions on the torus

A homeomorphism of the $2$-torus with Lefschetz number different from zero has a fixed point. We give a version of this result for nilpotent groups of diffeomorphisms. We prove that a nilpotent group of $2$-torus diffeomorphims has finite orbits when the group has some element with Lefschetz number different from zero.