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

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Let X be a compact differentiable manifold. A \(C^1\) map \(f:X\rightarrow X\) is called transversal if for all positive integers m, the graph of \(f^m\) intersects transversally the diagonal of \(X\times X\) at (xx) for any x fixed point of \(f^m\). In the present article, we describe the periodic structure of transversal maps on the n-dimensional torus. In particular, we give conditions on the eigenvalues of the induced linear map on the first homology, in order that all sufficiently large odd numbers are periods of the map. We present similar results for transversal maps on products of spheres of the same dimension. Later we generalize these results for transversal self-maps on rational exterior spaces of rank n.

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  1. 1.

    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)

  2. 2.

    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)

  3. 3.

    Brown, R.F.: The Lefschetz Fixed Point Theorem. Scott, Foresman and Company, Glenview (1971)

  4. 4.

    Casasayas, J., Llibre, J., Nunes, A.: Periodic orbits of transversal maps. Math. Proc. Camb. Philos. Soc. 118, 161–181 (1995)

  5. 5.

    Dold, A.: Fixed point indices of iterated maps. Invent. Math. 74, 419–435 (1983)

  6. 6.

    Duan, H.: The Lefschetz numbers of iterated maps. Topol. Appl. 67(1), 71–79 (1995)

  7. 7.

    Fagella, N., Llibre, J.: Periodic points of holomorphic maps via Lefschetz numbers. Trans. Am. Math. Soc. 352, 4711–4730 (2000)

  8. 8.

    Franks, J.: Some smooth maps with infinitely many hyperbolic periodic points. Trans. Am. Math. Soc. 226, 175–179 (1977)

  9. 9.

    Franks, J.: Period doubling and the Lefschetz formula. Trans. Am. Math. Soc. 287, 275–283 (1985)

  10. 10.

    García Guirao, J.L., Llibre, J.: Periodic structure of transversal maps on \({\mathbb{C}} P^n\), \({\mathbb{H}} P^n\) and \({\mathbb{S}}^p\times {\mathbb{S}}^q\). Qual. Theory Dyn. Syst. 12(2), 417–425 (2013)

  11. 11.

    Graff, G.: Minimal periods of maps of rational exterior spaces. Fund. Math. 163, 99–115 (2000)

  12. 12.

    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)

  13. 13.

    Jezierski, J., Marzantowicz, W.: Homotopy Methods in Topological Fixed and Periodic Points Theory. Springer, Berlin (2006)

  14. 14.

    Li, T.Y., Yorke, J.: Period three implies chaos. Am. Math. Mon. 82, 985–992 (1975)

  15. 15.

    Lefschetz, S.: Intersections and transformations of complexes and manifolds. Trans. Am. Math. Soc. 28, 1–49 (1926)

  16. 16.

    Llibre, J.: Lefschetz numbers for periodic points. Contemp. Math. 152, 215–227 (1993). (Amer. Math. Soc., Providence)

  17. 17.

    Llibre, J.: A note on the set of periods of transversal homological sphere self-maps. J. Difference Equ. Appl. 9, 417–422 (2003)

  18. 18.

    Llibre, J., Paraños, J., Rodríguez, J.A.: Periods for transversal maps on compact manifolds with a given homology. Houston J. Math. 24, 397–407 (1998)

  19. 19.

    Llibre, J., Sirvent, V.F.: Partially periodic point free self-maps on surfaces, graphs, wedge sums and product of spheres. J. Difference Equ. Appl. 19, 1654–1662 (2013)

  20. 20.

    Llibre, J., Sirvent, V.F.: Periodic structure of transversal maps on sum-free products of spheres. J. Difference Equ. Appl. 25, 619–629 (2019)

  21. 21.

    Llibre, J., Swanson, R.: Periodic points for transversal maps on surfaces. Houston J. Math. 19, 395–403 (1993)

  22. 22.

    Matsuoka, T.: The number of periodic points of smooth maps. Ergodic Theory Dyn. Syst. 9, 153–163 (1989)

  23. 23.

    Matsuoka, T., Shiraki, H.: Smooth maps with finitely many periodic points. Mem. Fac. Sei. Kochi. Univ. (Math.) 11, 1–6 (1990)

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The author thanks the anonymous referees for useful suggestions that made possible to improve the quality and readability of the article.

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Correspondence to Víctor F. Sirvent.

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Sirvent, V.F. A Note on the Periodic Structure of Transversal Maps on the Torus and Products of Spheres. Qual. Theory Dyn. Syst. 19, 45 (2020).

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  • Transversal maps
  • Lefschetz numbers
  • Periodic point
  • Product of spheres
  • Torus
  • Rational exterior spaces

Mathematics Subject Classification

  • 37C25
  • 37C30
  • 37E15