A criterion for the existence of periodic points based on the eigenvalues of maps induced in cohomology

• Michalina Horecka ^[1] ; Paweł Raźny ^[1]
  1. [1] Jagiellonian University

     Jagiellonian University

     Kraków, Polonia

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 2, 2022
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We present a criterion for the existence of periodic points based on the eigenvalues of maps induced in cohomology for spaces with rational cohomology isomorphic to a tensor product of a graded exterior algebra with generators in odd dimensions and a graded algebra with all elements of even degree. We give a number of natural examples of such spaces and provide some non-trivial ones. We also give a counterexample to a claim in [3] given there without proof.

• Referencias bibliográficas
  • 1. Berrizbeitia, P., González, M.J., Sirvent, V.F.: On the Lefschetz zeta function and the minimal sets of Lefschetz periods for Morse-Smale...
  • 2. Chern, S.-S., Spanier, E.: The homology structure of sphere bundles. Proc. Nat. Acad. Sci. U.S.A. 36, 248–255 (1950)
  • 3. Duan, H.: The Lefschetz numbers of iterated maps. Topol Appl 67(1), 71–79 (1995)
  • 4. Franks, J.: Homology and dynamical systems. CBMS Regional Conf. Ser. in Math. 49, Providence, Rhode Island (1982)
  • 5. Graff, G.: Minimal periods of maps of rational exterior spaces. Fund. Math. 163(2), 99–115 (2000)
  • 6. Graff, G., Kaczkowska, A., Nowak-Przygodzki, P., Signerska, J.: Lefschetz periodic point free selfmaps of compact manifolds. Topology Appl....
  • 7. Guirao, J., Llibre, J.: On the Lefschetz periodic point free continuous self-maps on connected compact manifolds. Topology Appl. 158(16),...
  • 8. Hatcher, A.: Algebraic topology. Cambridge University Press, Cambridge (2002)
  • 9. Llibre, J., Sirvent, V.F.: On Lefschetz periodic point free self-maps. J. Fix Point Theory A. 20, 38 (2018). https://doi.org/10.1007/s11784-018-0498-5
  • 10. Llibre, J., Sirvent, V.F.: Partially periodic point free self-maps on surfaces, graphs, wedge sums and products of spheres. J. Dif. Equ....
  • 11. Sirvent, V.F.: Partially periodic point free self-maps on product of spheres and Lie groups. Qual. Theory Dyn. Syst. 19(3), 1–12 (2020)
  • 12. Thurston, W.P.: Some simple examples of symplectic manifolds. Proc. Amer. Math. Soc. 55, 467–468 (1976)