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The Anosov relation for Nielsen numbers of maps of infra-nilmanifolds. (English) Zbl 1112.55002

The authors prove several theorems on the relation between the Lefschetz number \(L(f)\) and the Nielsen number \(N(f)\) for continuous maps on infra-nilmanifolds. More specifically, let \(M\) be an infra-nilmanifold and \(f:M\to M\) continuous. If the holonomy group of \(M\) is of odd order then \(N(f)=| L(f)| \). Suppose that \(M\) is closed and smooth and \(f\) is a \(C^1\)-map. Then \(f\) is called expanding if for some Riemannian metric there are \(C>0\) and \(\mu>1\) such that \(\| Df^n(v)\| \geq C\mu^n\| v\| \) for all \(v\in TM\). The authors prove that \(N(f)=| L(f)| \) for an expanding map if and only if \(M\) is orientable. An infra-nilmanifold is an orbit space \(M=E\backslash G\) where \(G\) is a connected, simply connected, nilpotent Lie group and \(E\) is a torsion-free almost-crystallographic group. If \(f:M\to M\) is continuous then \(f\) is homotopic to a map which is induced by an affine endomorphism \((\delta,\mathcal{D}):G\to G\). Then \(f\) is called nowhere expanding if all eigenvalues of \(\mathcal{D}_*\) have modulus at most 1. The authors prove that \(N(f)=L(f)\) for nowhere expanding maps \(f:M\to M\).

MSC:

55M20 Fixed points and coincidences in algebraic topology
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics

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References:

[1] Anosov D (1985) The Nielsen numbers of maps of nil-manifolds. Uspekhi Mat Nauk 40: 133–134. English transl: Russian Math Surveys 40: 149–150 · Zbl 0594.55002
[3] Brown RF (1971) The Lefschetz Fixed Point Theorem. Geenview, Icc. Scott, Foresman and Comp · Zbl 0216.19601
[4] Dekimpe K (1996) Almost-Bieberbach Groups: Affine and Polynomial Structures. Lect Notes Math 1639: Berlin Heidelberg New York: Springer · Zbl 0865.20001
[5] Dekimpe K, De Rock B, Malfait W (2005) The Anosov theorem for infra-nilmanifolds with cyclic holonomy group. Pacific J Math, to appear · Zbl 1098.55002
[7] Dekimpe K, De Rock B, Pouseele H (2006) The Anosov theorem for infra-nilmanifolds with an odd order abelian holonomy group. Fixed Point Theory and Applications, Article ID 63939, 12 p · Zbl 1113.55001
[10] Jiang B (1983) Nielsen Fixed Point Theory. Providence, RI: Amer Math Soc · Zbl 0512.55003
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