On the Lefschetz zeta function for quasi-unipotent maps on the \(n\)-dimensional torus. (English) Zbl 1305.37021

Berrizbeitia, Pedro; Sirvent, Víctor F.

J. Difference Equ. Appl. 20, No. 7, 961-972 (2014).

As mentioned in the paper, a continuous self map \(f\) on the \(n\)-dimensional torus \(T^{n}\) is called quasi-unipotent if for each \(0\leq k \leq n\), the eigenvalues of the self map \(f_{*k}\) on the \(k\)th rational homology group \(H_{k}(T^{n},Q)\) are root of unity. “The Lefschetz zeta function of \(f\) is \(\zeta_{f}(t)=\exp(\Sigma_{m\geq 1}\frac{L(f^{m})}{m}t^{m})\), where \(L(f)=\Sigma_{k=0}^{n}(-1)^{k}\mathrm{trace}(f_{*k}).\)” In this interesting paper the authors compute \(\zeta_{f}(t)\) for a quasi-unipotent map on \(T^{n}\) when \(n=p-1\) and \(p\) is an odd prime. In fact they show that \(\zeta_{f}(t)=\pm c_{1}(t)^{1-p}c_{p}(t)\), where \(c_{p}(t)\) is the characteristic polynomial of \(f_{*1}\). They characterize “the minimal set of Lefschetz periods for \(C^{1}\) quasi-unipotent maps of \(T^{n}\) having finitely many periodic points, all of them hyperbolic.”

Reviewer: Mohammad Reza Molaei (Kerman)

MSC:

37E15	Combinatorial dynamics (types of periodic orbits)
37D15	Morse-Smale systems
37C25	Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
11T22	Cyclotomy

Keywords:

cyclotomic polynomial; Lefschetz zeta function; periodic point; minimal set of Lefschetz periods; Morse-Smale diffeomorphisms; quasi-unipotent map

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