Abstract
In the present article we study the periodic structure of a class of maps on the n-dimensional torus such that the eigenvalues of the induced map on the first homology are dilations of roots of unity. This family of maps shares some properties with the family of the quasi-unipotent maps. We compute their Lefschetz numbers, and show that the Lefschetz numbers of period m are non-zero, for all m’s, in the case that n is an odd prime. Moreover we give an explicit expression for their corresponding Lefschetz zeta functions. We conjecture that in any dimension the Lefschetz numbers of period m are non-zero, for all m.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)
Artin, M., Mazur, B.: On periodic points. Ann. Math. 81, 82–99 (1965)
Baake, M., Hermisson, J., Pleasants, P.A.B.: The torus parametrization of quasiperiodic LI-classes. J. Phys. A Math. Gen. 30, 3029–3056 (1997)
Baake, M., Lau, E., Paskunas, V.: A note on the dynamical zeta function of general toral automorphisms. Monatsh. Math. 161, 33–42 (2010)
Babenko, I.K., Bogatyi, S.A.: The behaviour of the index of periodic points under iterations of a mapping. Math. USSR Izv. 38, 1–26 (1992)
Banchoff, T., Rosen, M.I.: Periodic points of Anosov diffeomorphisms. 1970 Global Analysis Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., pp. 17–21. Amer. Math. Soc., Providence, RI (1968)
Berrizbeitia, P., Sirvent, V.F.: On the Lefschetz zeta function for quasi-unipotent maps on the \(n\)-dimensional torus. J. Differ. Equ. Appl. 20, 961–972 (2014)
Berrizbeitia, P., González, M.J., Mendoza, A., Sirvent, V.F.: On the Lefschetz zeta function for quasi-unipotent maps on the\(n\)-dimensional torus II: the general case. Topol. Appl. 210, 246–262 (2016)
Berrizbeitia, P., González, M.J., Sirvent, V.F.: On the Lefschetz zeta function and the minimal sets of Lefschetz periods for Morse-Smale diffeomorphisms on products of \(\ell \)-spheres. Topol. Appl. 235, 428–444 (2018)
Byszewski, J., Graff, G., Ward, T.: Dold sequences, periodic points, and dynamics. arXiv:2007.04031
Dold, A.: Fixed point indices of iterated maps. Invent. math. 74, 419–435 (1983)
Dos Santos, N.M., Urzúa, R.: Minimal homeomorphisms on low-dimensional tori. Ergod. Theory. Dyn. Syst. 29, 1515–1528 (2009)
Duan, H.: The Lefschetz numbers of iterated maps. Topol. Appl. 67, 71–79 (1995)
Franks, J.: Period doubling and the Lefschetz formula. Trans. Am. Math. Soc. 287, 275–283 (1985)
Graff, G.: Minimal periods of maps of rational exterior spaces. Fund. Math. 163, 99–115 (2000)
Graff, G., Lebiedź, M., Myszkowski, A.: Periodic expansion in determining minimal sets of Lefschetz periods for Morse-Smale diffeomorphisms. J. Fixed Point Theory Appl. 21(47), 21 (2019)
Graff, G., Lebieź, M., Nowak-Przygodzki, P.: Generating sequences of Lefschetz numbers of iterates. Monatsh. Math. 188, 511–525 (2019)
Guillamón, A., Jarque, X., Llibre, J., Ortega, J., Torregrosa, J.: Periods for transversal maps via Lefschetz numbers for periodic points. Trans. Math. Soc. 347, 4779–4806 (1995)
Guirao, J.L.G., Llibre, J.: Periods of Morse-Smale diffeomorphisms of \({\mathbb{S}}^2\). Colloq. Math. 1(10), 477–483 (2008)
Guirao, J.L.G., Llibre, J.: Minimal Lefschetz sets of periods for Morse-Smale diffeomorphisms on the \(n\)-dimensional torus. J. Differ. Equ. Appl. 16, 689–703 (2010)
Guirao, J.L.G., Llibre, J.: The set of periods for the Morse-Smale diffeomorphisms on \({\mathbb{T}}^2\). Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 19, 471–484 (2012)
Guirao, J.L.G., Llibre, J.: On the set of periods for the Morse-Smale diffeomorphisms on the disc with \(N\)-holes. J. Differ. Equ. Appl. 19, 1161–1173 (2013)
Jezierski, J., Marzantowicz, W.: Homotopy Methods in Topological Fixed and Periodic Points Theory. Springer, Berlin (2006)
Lang, S.: Algebra, Third Edition, Graduate Texts in Mathematics 211. Springer, Berlin (2005)
Lefschetz, S.: Intersections and transformations of complexes and manifolds. Trans. Am. Math. Soc. 28, 1–49 (1926)
Llibre, J.: Lefschetz numbers for periodic points. Contemp. Math. 152, 215–227 (1993)
Llibre, J., Sirvent, V.F.: Minimal sets of periods for Morse-Smale diffeomorphisms on orientable compact surfaces. Houst. J. Math. 35, 835–855 (2009)
Llibre, J., Sirvent, V.F.: A survey on the minimal sets of Lefschetz periods for Morse-Smale diffeomorphisms on some closed manifolds. Publicaciones Matemáticas del Uruguay 14, 155–169 (2013)
Metropolis, N., Rota, G.: Witt vectors and the algebra of necklaces. Adv. Math. 50(2), 95–125 (1983)
Rotman, J.J.: An Introduction to Algebraic Topology. GTM, Springer, Berlin (1993)
Shub, M., Sullivan, D.: Homology theory and dynamical systems. Topology 14, 109–132 (1975)
Sirvent, V.F.: A note on the periodic structure of transversal maps on the torus and the product of spheres. Qual. Theory Dyn. Syst. 19, 45 (2020)
Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73(6), 747–817 (1967)
Vick, J.W.: Homology Theory. An Introduction to Algebraic Topology. Springer, New York (1994)
Walters, P.: An Introduction to Ergodic Theory. Springer, New York (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Berrizbeitia, P., González, M.J. & Sirvent, V.F. On the Lefschetz Zeta Function for a Class of Toral Maps. Qual. Theory Dyn. Syst. 20, 17 (2021). https://doi.org/10.1007/s12346-021-00453-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-021-00453-1