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An averaging formula for Nielsen numbers of affine n-valued maps on infra-nilmanifolds

Karel Dekimpe, Lore De Weerdt

Abstract

In [8,9], the authors developed a nice formula to compute the Nielsen number of a self-map on an infra-nilmanifold. For the case of nilmanifolds this formula was extended to $n$-valued maps in [4]. In this paper, we extend these results further and establish the averaging formula to compute the Nielsen number of any $n$-valued affine map on an infra-nilmanifold.

An averaging formula for Nielsen numbers of affine n-valued maps on infra-nilmanifolds

Abstract

In [8,9], the authors developed a nice formula to compute the Nielsen number of a self-map on an infra-nilmanifold. For the case of nilmanifolds this formula was extended to -valued maps in [4]. In this paper, we extend these results further and establish the averaging formula to compute the Nielsen number of any -valued affine map on an infra-nilmanifold.
Paper Structure (5 sections, 10 theorems, 60 equations)

This paper contains 5 sections, 10 theorems, 60 equations.

Table of Contents

  1. Nielsen theory of $n$-valued maps
  2. Infra-nilmanifolds and affine maps
  3. Decomposition of the fixed point set
  4. Averaging formula for affine maps
  5. Example

Key Result

Theorem 2.1

Let $N\backslash G$ be a nilmanifold. The Nielsen number of an affine map $f:N\backslash G\to D_n(N\backslash G)$ with lift $\tilde{f}:G\to F_n(G,N):x\mapsto (g_1\varphi_1(x),\ldots,g_n\varphi_n(x))$ is given by

Theorems & Definitions (19)

  • Remark 1.1
  • Theorem 2.1: charlotte2
  • Lemma 3.1: penninckx
  • Lemma 4.1
  • proof
  • Lemma 4.2: charlotte2
  • Corollary 4.4
  • proof
  • Lemma 4.5
  • proof
  • ...and 9 more