Ir al contenido

Documat

Dold Coefficients of Quasi-unipotent Homeomorphisms of Orientable Surfaces

  • Grzegorz Graff [1] ; Wacław Marzantowicz [2] ; Łukasz Patryk Michalak [3]
    1. [1] University of Technology

      University of Technology

      Rusia

    2. [2] Adam Mickiewicz University of Poznan
    3. [3] Institute of Mathematics, Physics and Mechanics, Jadranska
    Mostrar afiliaciones +
  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 3, 2025
  • Idioma: inglés
  • Enlaces
  • Resumen

    • The sequence of Dold coefficients (an( f )) of a self-map f : X → X forms a dual sequence to the sequence of Lefschetz numbers (L( f n)) of iterations of f under the Möbius inversion formula. The set AP( f ) = {n : an( f ) = 0} is called the set of algebraic periods of f . Both the set of algebraic periods and sequence of Dold coefficients play an important role in dynamical systems and periodic point theory. In this work we provide a description of surface homeomorphisms with bounded (L( f n)) (quasi-unipotent maps) in terms of Dold coefficients. We also discuss the problem of minimization of the genus of a surface for which one can realize a given set of natural numbers as the set of algebraic periods. Finally, we compute and list all possible Dold coefficients and algebraic periods for a given orientable surface with small genus and give some geometrical applications of the obtained results.

  • Referencias bibliográficas
    • 1. Babenko, I.K., Bogaty˘ı, S.A.: Behavior of the index of periodic points under iterations of a mapping. (Russian) Izv. Akad. Nauk SSSR Ser....
    • 2. Berrizbeitia, P., González, M., Sirvent, V.: On the Lefschetz zeta function and the minimal sets of Lefschetz periods for Morse–Smale diffeomorphisms...
    • 3. Byszewski, J., Graff, G., Ward, T.: Dold sequences, periodic points, and dynamics. Bull. Lond. Math. Soc. 53(5), 1263–1298 (2021)
    • 4. Cufí-Cabré, C., Llibre, J.: Periods of Morse–Smale diffeomorphisms on Sn, Sm × Sn, CPn and CPn. J. Fixed Point Theory Appl. 24, 4 (2022)
    • 5. Dicks, W., Llibre, J.: Orientation-preserving self-homeomorphisms of the surface of genus two have points of period at most two. Proc....
    • 6. Dold, A.: Fixed point indices of iterated maps. Invent. Math. 74, 419–435 (1983)
    • 7. Duan, H., Graff, G., Jezierski, J., Myszkowski, A.: Algebraic periods and minimal number of periodic points for smooth self-maps of 1-connected...
    • 8. Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Mathematical Series, 49. Princeton University Press, Princeton (2012)
    • 9. Franks, J.: Some smooth maps with infinitely many hyperbolic points. Trans. Am. Math. Soc. 226, 175–179 (1977)
    • 10. Grines, V., Medvedev, T., Pochinka, O.: Dynamical Systems on 2- and 3-Manifolds. Springer, Berlin (2018)
    • 11. Graff, G.: Minimal periods of maps of rational exterior spaces. Fund. Math. 163(2), 99–115 (2000)
    • 12. Graff, G., Lebied´z, M., Myszkowski, A.: Periodic expansion in determining minimal sets of Lefschetz periods for Morse–Smale diffeomorphisms....
    • 13. Graff, G., Lebied´z, M., Nowak-Przygodzki, P.: Generating sequences of Lefschetz numbers of iterates. Monatsh. Math. 188(3), 511–525 (2019)
    • 14. Graff, G., Marzantowicz, W., Michalak, Ł.P.: Database of algebraic periods of quasi-unipotent orientation-preserving homeomorphisms of...
    • 15. Graff, G., Marzantowicz, W., Michalak, Ł.P., Myszkowski, A.: Every finite set of natural numbers is realizable as algebraic periods of...
    • 16. Guirao, J.L., Llibre, J.: Periodic structure of transversal maps on C Pn, H Pn and S p × Sq . Qual. Theory Dyn. Syst. 12(2), 417–425 (2013)
    • 17. Guirao, J.L., Llibre, J.: Periods ofMorse–Smale diffeomorphisms of Sn. Colloq.Math. 110(2), 477–483 (2008)
    • 18. Guirao, J.L., Llibre, J.: Minimal Lefschetz sets of periods for Morse–Smale diffeomorphisms on the n-dimensional torus. J. Differ. Equ....
    • 19. Gupta, H.: Euler’s totient function and its inverse. Indian J. Pure Appl. Math. 12(1), 22–30 (1981)
    • 20. Jezierski, J., Marzantowicz, W.: Homotopy Methods in Topological Fixed and Periodic Points Theory. Topological Fixed Point Theory and...
    • 21. Jiang, B.: Estimation of the number of periodic orbits. Pac. J. Math. 172(1), 151–185 (1996)
    • 22. Llibre, J.: Lefschetz Numbers for Periodic Points. Contemporary Mathematics, pp. 215–227. Amer. Math. Soc., Providence (1993)
    • 23. Llibre, J., Paranõs, J., Rodriguez, J.A.: Periods for transversal maps on compact manifolds with a given homology. Houst. J. Math. 24,...
    • 24. Llibre, J., Sirvent, V.F.: Minimal sets of periods for Morse–Smale diffeomorphisms on orientable compact surfaces. Houst. J. Math. 35(3),...
    • 25. Llibre, J., Sirvent, V.F.: A survey on the minimal sets of Lefschetz periods for Morse–Smale diffeomorphisms on some closed manifolds,...
    • 26. Llibre, J., Sirvent, V.F.: Minimal sets of periods for Morse–Smale diffeomorphisms on non-orientable compact surfaces without boundary....
    • 27. Marzantowicz, W., Nowak-Przygodzki, P.: Finding periodic points of a map by use of a k-adic expansion. Discrete Contin. Dyn. Syst. 5(3),...
    • 28. Myszkowski, A.: Ph.D. Thesis, Gda ´nsk University of Technology, (in Polish) (2023)
    • 29. Newman, M.: Integral Matrices, Pure and Applied Mathematics, vol. 45. Academic Press, New York (1972)
    • 30. Nielsen, J.: Fixpunktfrie Afbildninger. Mat. Tidsskr. B 25–41 (1942)
    • 31. Nielsen, J.: Surface transformation classes of algebraically finite type. Danske Vid. Selsk. Mat.-Fys. Medd. 21(2), 89 (1944)
    • 32. da Rocha, L.F.: Characterization of Morse–Smale isotopy classes on surfaces. Ergod. Theory Dyn. Syst. 5, 107–122 (1985)
    • 33. Seidel, P.: Trasversalität für periodische Punkte differenziebarer Abbildungen,Mathematisches Institut, Universität Heidelberg, Forschergruppe:...
    • 34. Stanley, R.: Enumerative combinatorics, vol. I. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey (1986)
    • 35. Shub, M.: Morse–Smale Diffeomorphisms are Unipotent on Homology, Dynamical systems, (Proc. Sympos., Univ. Bahia, Salvador). Academic Press...
    • 36. Sirvent, V.: On the periodic structure of self-maps on the product of spheres of different dimensions. J. Fixed Point Theory Appl. 26,...
    • 37. Yang, Q.: Conjugacy classes in integral symplectic groups. Linear Algebra Appl. 418, 614–624 (2006)
    • 38. Yang, Q.: Decomposability of symplectic matrices over principal ideal domain. J. Number Theory 149, 139–152 (2015)
Fundación Dialnet

Mi Documat

Opciones de artículo

Opciones de compartir

Opciones de entorno