On the Lefschetz Zeta Function for a Class of Toral Maps

• Berrizbeitia, Pedro ^[2] ; González, Marcos J. ^[2] ; Sirvent, Víctor F. ^[1]
  1. [1] Universidad Católica del Norte

     Universidad Católica del Norte

     Antofagasta, Chile

     
  2. [2] Universidad Simón Bolívar
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 1, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00453-1
• Enlaces
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• Resumen

  • In the present article we study the periodic structure of a class of maps on the ndimensional 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.

• Referencias bibliográficas
  • 1. Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)
  • 2. Artin, M., Mazur, B.: On periodic points. Ann. Math. 81, 82–99 (1965)
  • 3. Baake, M., Hermisson, J., Pleasants, P.A.B.: The torus parametrization of quasiperiodic LI-classes. J. Phys. A Math. Gen. 30, 3029–3056...
  • 4. Baake, M., Lau, E., Paskunas, V.: A note on the dynamical zeta function of general toral automorphisms. Monatsh. Math. 161, 33–42 (2010)
  • 5. 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)
  • 6. Banchoff, T., Rosen, M.I.: Periodic points of Anosov diffeomorphisms. 1970 Global Analysis Proc. Sympos. Pure Math., Vol. XIV, Berkeley,...
  • 7. Berrizbeitia, P., Sirvent, V.F.: On the Lefschetz zeta function for quasi-unipotent maps on the ndimensional torus. J. Differ. Equ. Appl....
  • 8. Berrizbeitia, P., González, M.J., Mendoza, A., Sirvent, V.F.: On the Lefschetz zeta function for quasiunipotent maps on then-dimensional...
  • 9. Berrizbeitia, P., González, M.J., Sirvent, V.F.: On the Lefschetz zeta function and the minimal sets of Lefschetz periods for Morse-Smale...
  • 10. Byszewski, J., Graff, G., Ward, T.: Dold sequences, periodic points, and dynamics. arXiv:2007.04031
  • 11. Dold, A.: Fixed point indices of iterated maps. Invent. math. 74, 419–435 (1983)
  • 12. Dos Santos, N.M., Urzúa, R.: Minimal homeomorphisms on low-dimensional tori. Ergod. Theory. Dyn. Syst. 29, 1515–1528 (2009)
  • 13. Duan, H.: The Lefschetz numbers of iterated maps. Topol. Appl. 67, 71–79 (1995)
  • 14. Franks, J.: Period doubling and the Lefschetz formula. Trans. Am. Math. Soc. 287, 275–283 (1985)
  • 15. Graff, G.: Minimal periods of maps of rational exterior spaces. Fund. Math. 163, 99–115 (2000)
  • 16. Graff, G., Lebied´z, M., Myszkowski, A.: Periodic expansion in determining minimal sets of Lefschetz periods for Morse-Smale diffeomorphisms....
  • 17. Graff, G., Lebie´z, M., Nowak-Przygodzki, P.: Generating sequences of Lefschetz numbers of iterates. Monatsh. Math. 188, 511–525 (2019)
  • 18. Guillamón, A., Jarque, X., Llibre, J., Ortega, J., Torregrosa, J.: Periods for transversal maps via Lefschetz numbers for periodic points....
  • 19. Guirao, J.L.G., Llibre, J.: Periods of Morse-Smale diffeomorphisms of S2. Colloq. Math. 1(10), 477–483 (2008)
  • 20. Guirao, J.L.G., Llibre, J.: Minimal Lefschetz sets of periods for Morse-Smale diffeomorphisms on the n-dimensional torus. J. Differ. Equ....
  • 21. Guirao, J.L.G., Llibre, J.: The set of periods for the Morse-Smale diffeomorphisms on T2. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math....
  • 22. 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....
  • 23. Jezierski, J., Marzantowicz, W.: Homotopy Methods in Topological Fixed and Periodic Points Theory. Springer, Berlin (2006)
  • 24. Lang, S.: Algebra, Third Edition, Graduate Texts in Mathematics 211. Springer, Berlin (2005)
  • 25. Lefschetz, S.: Intersections and transformations of complexes and manifolds. Trans. Am. Math. Soc. 28, 1–49 (1926)
  • 26. Llibre, J.: Lefschetz numbers for periodic points. Contemp. Math. 152, 215–227 (1993)
  • 27. Llibre, J., Sirvent, V.F.: Minimal sets of periods for Morse-Smale diffeomorphisms on orientable compact surfaces. Houst. J. Math. 35,...
  • 28. Llibre, J., Sirvent, V.F.: A survey on the minimal sets of Lefschetz periods for Morse-Smale diffeomorphisms on some closed manifolds....
  • 29. Metropolis, N., Rota, G.: Witt vectors and the algebra of necklaces. Adv. Math. 50(2), 95–125 (1983)
  • 30. Rotman, J.J.: An Introduction to Algebraic Topology. GTM, Springer, Berlin (1993)
  • 31. Shub, M., Sullivan, D.: Homology theory and dynamical systems. Topology 14, 109–132 (1975)
  • 32. 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,...
  • 33. Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73(6), 747–817 (1967)
  • 34. Vick, J.W.: Homology Theory. An Introduction to Algebraic Topology. Springer, New York (1994)
  • 35. Walters, P.: An Introduction to Ergodic Theory. Springer, New York (2000)