On the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus. II: The general case

Berrizbeitia, Pedro; González, Marcos J.; Mendoza, Alberto; Sirvent, Vı́ctor F.

doi:10.1016/j.topol.2016.07.020

Cited by 7 publications

(4 citation statements)

References 20 publications

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“…In the case of quasi-unipotent maps of tori the Lefschetz zeta function can be computed completely explicitly. Indeed, it is shown in [7], [8] that, for a quasiunipotent self map f : T n R → T n R , the Lefschetz zeta function has an explicit form that is completely determined by the map on the first homology. Under the quasiunipotent assumption all the eigenvalues of the induced map on H 1 are roots of unity, hence the characteristic polynomial det(1…”

Section: 1mentioning

confidence: 99%

Bost-Connes systems and F1-structures in Grothendieck rings, spectra, and Nori motives

F.¹,

Manin²,

Marcolli³

2019

Preprint

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We construct geometric lifts of the Bost-Connes algebra to Grothendieck rings and to the associated assembler categories and spectra, as well as to certain categories of Nori motives. These categorifications are related to the integral Bost-Connes algebra via suitable Euler characteristic type maps and zeta functions, and in the motivic case via fiber functors. We also discuss aspects of F 1 -geometry, in the framework of torifications, that fit into this general setting.

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Section: 1mentioning

confidence: 99%

Bost-Connes systems and F1-structures in Grothendieck rings, spectra, and Nori motives

F.¹,

Manin²,

Marcolli³

2019

Preprint

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“…Furthermore if the eigenvalues of f * 1 are root of unity then the Lefschetz zeta function of f is of the form (cf. [3]):…”

Section: The N-dimensional Torus: T Nmentioning

confidence: 99%

“…or ζ f (t) = 1, with positive integers d i and integers r i . Moreover in [3] there are explicit formulae for the values of d i and r i in terms of the characteristic polynomial of f * 1 .…”

Section: The N-dimensional Torus: T Nmentioning

confidence: 99%

On topological entropy, Lefschetz numbers and Lefschetz zeta functions

Llibre

Sirvent

2019

Topology and its Applications

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“…In the particular situation of the torus, i.e. n = 1, the study was done previously in [2,13]. Regarding the periodic structure of transversal maps for X = X(n, s), for n ≥ 1 and s > 1, see [30].…”

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confidence: 99%

A Note on the Periodic Structure of Transversal Maps on the Torus and Products of Spheres

Sirvent

2020

Qual. Theory Dyn. Syst.

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In the present article we study the periodic structure of some well-known classes of C 1 self-maps on the product of spheres of different dimensions: transversal maps, Morse-Smale diffeomorphisms and maps with all its periodic points hyperbolic. Our approach is via the Lefschetz fixed point theory. We give a complete characterization of the minimal set of Lefschetz periods for Morse-Smale diffeomorphisms on these spaces. We also consider C 1 maps with all its periodic points hyperbolic and we give conditions for these maps to have infinitely many periodic points. We describe the period set of the transversal maps on these spaces. Finally we applied these results to describe the periodic structure of similar classes of maps on compact, connected, simple connected and simple Lie groups.

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On the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus. II: The general case

Cited by 7 publications

References 20 publications

Bost-Connes systems and F1-structures in Grothendieck rings, spectra, and Nori motives

Bost-Connes systems and F1-structures in Grothendieck rings, spectra, and Nori motives

On topological entropy, Lefschetz numbers and Lefschetz zeta functions

A Note on the Periodic Structure of Transversal Maps on the Torus and Products of Spheres

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