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Strong Nielsen equivalence on the punctured disc

Stavroula Makri

Abstract

Let $f$ be an orientation-preserving homeomorphism of the 2-disc $\mathbb{D}^2$ that fixes the boundary pointwise and leaves invariant a finite subset in the interior of $\mathbb{D}^2$. We study the strong Nielsen equivalence of periodic points of such homeomorphisms $f$ and we give a necessary and sufficient condition for two periodic points to be strong Nielsen equivalent in the context of braid theory. In addition, we present an application of our result to the trace formula given by Jiang--Zheng, deducing that the obtained forced periodic orbits belong to different strong Nielsen classes.

Strong Nielsen equivalence on the punctured disc

Abstract

Let be an orientation-preserving homeomorphism of the 2-disc that fixes the boundary pointwise and leaves invariant a finite subset in the interior of . We study the strong Nielsen equivalence of periodic points of such homeomorphisms and we give a necessary and sufficient condition for two periodic points to be strong Nielsen equivalent in the context of braid theory. In addition, we present an application of our result to the trace formula given by Jiang--Zheng, deducing that the obtained forced periodic orbits belong to different strong Nielsen classes.
Paper Structure (5 sections, 8 theorems, 4 equations, 1 figure)

This paper contains 5 sections, 8 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Remarks
  5. Braid forcing

Key Result

Proposition 1.1

Let $f:M\to M$ be an orientation-preserving homeomorphism and let $x,y\in P_n(f)$. It holds that $x,y$ are strong Nielsen equivalent, denoted by $x \mathrel{\overset{\text{\tiny SN}}{\hbox{[}1]{$∼$}}} y$, if and only if $snt(x,f)=snt(y,f)$.

Figures (1)

  • Figure 1: Two orbits of period two that are periodic Nielsen equivalent but not strong Nielsen equivalent.

Theorems & Definitions (26)

  • Proposition 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 16 more