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Uniqueness of dynamical zeta functions and symmetric products

Abstract

A characterization of dynamically defined zeta functions is presented. It comprises a list of axioms, natural extension of the one which characterizes topological degree, and a uniqueness theorem. Lefschetz zeta function is the main (and proved unique) example of such zeta functions. Another interpretation of this function arises from the notion of symmetric product from which some corollaries and applications are obtained.

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References

  1. 1.

    Amann H., Weiss S.: On the uniqueness of the topological degree. Math. Z. 130, 39–54 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Borsuk K., Ulam S.: On symmetric products of topological spaces. Bull. Amer. Math. Soc. 37, 875–882 (1931)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Curtis D. W.: Growth hyperspaces of Peano continua. Trans. Amer. Math. Soc. 238, 271–283 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Curtis D. W., Schori R. M.: Hyperspaces of polyedra are Hilbert cubes. Fund. Math. 99, 189–197 (1978)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Dold A.: Homology of symmetric products and other functors of complexes. Ann. of Math. (2) 68, 54–80 (1958)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Dold A.: Fixed point index and fixed point theorem for Euclidean neighborhood retracts. Topology 4, 1–8 (1965)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Dold A.: Lectures on Algebraic Topology. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  8. 8.

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

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Dold A.: Combinatorial and geometric fixed point theory. Riv. Mat. Univ. Parma (4) 10, 23–32 (1984)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Dold A., Thom R.: Quasifaserungen und unendliche symmetrische Produkte. Ann. of Math. (2) 67, 239–281 (1958)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    J. Dugundji and A. Granas, Fixed Point Theory. Monografie Matematyczne, PWN-Polish Scientific Publishers, Warsaw, 1982.

  12. 12.

    Floyd E.: Orbit spaces of finite transformation groups. II. Duke Math. J. 22, 33–38 (1955)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    L. Führer, Theorie des Abbildungsgrades in endlichdimensionalen Räumen. Dissertation, Freie Univ. Berlin, Berlin, 1972.

  14. 14.

    Górniewicz L.: Topological Fixed Point Theory of Multivalued Mappings. Springer, Dordrecht (2006)

    MATH  Google Scholar 

  15. 15.

    Jaworowski J.: Symmetric products of ANR’s associated with a permutation group. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20, 649–651 (1972)

    MathSciNet  MATH  Google Scholar 

  16. 16.

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

    MATH  Google Scholar 

  17. 17.

    Le Calvez P.: Dynamique des homéomorphismes du plan au voisinage d’un point fixe. Ann. Sc. Éc. Norm. Supér. (4) 36, 139–171 (2003)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Liao S. D.: On the topology of cyclic products of spheres. Trans. Amer. Math. Soc. 77, 520–551 (1954)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Macdonald I. G.: The Poincar´e polynomial of a symmetric product. Proc. Cambridge Philos. Soc. 58, 563–568 (1962)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Masih S.: Fixed points of symmetric product mappings of polyhedra and metric absolute neighborhood retracts. Fund. Math. 80, 149–156 (1973)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Masih S.: On the fixed point index and the Nielsen fixed point theorem of symmetric product mappings. Fund. Math. 102, 143–158 (1979)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    McDuff D.: Configuration spaces of positive and negative particles. Topology 14, 91–107 (1975)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    W. de Melo, Topologia das Variedades. IMPA, http://w3.impa.br/~demelo/.

  24. 24.

    Rallis N.: A fixed point index theory for symmetric product mappings. Manuscripta Math. 44, 279–308 (1983)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.

  26. 26.

    Ruiz del Portal F. R., Salazar J. M.: Fixed point index in hyperspaces: A Conley-type index for discrete semidynamical systems. J. Lond. Math. Soc. (2) 64, 191–204 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Salamon D.: Seiberg-Witten invariants of mapping tori, symplectic fixed points, and Lefschetz numbers. Turkish J. Math. 23, 117–143 (1999)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    J. M. Salazar, Fixed point index in symmetric products. Trans. Amer. Math. Soc. 357, 3493–3005 (2005)

  29. 29.

    R. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Stud. Adv. Math. 62, Cambridge Univ. Press, Cambridge, 1999.

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Correspondence to Luis Hernández-Corbato.

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Blanco Gómez, E., Hernández-Corbato, L. & Ruiz del Portal, F.R. Uniqueness of dynamical zeta functions and symmetric products. J. Fixed Point Theory Appl. 18, 689–719 (2016). https://doi.org/10.1007/s11784-016-0296-x

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Mathematics Subject Classification

  • 37C30
  • 55M20

Keywords

  • Lefschetz zeta function
  • fixed point index
  • symmetric products