A Signed Version of Putnam’s Homology Theory: Lefschetz and Zeta Functions

  • Robin J. DeeleyEmail author
Part of the MATRIX Book Series book series (MXBS, volume 1)


A signed version of Putnam homology for Smale spaces is introduced. Its definition, basic properties and associated Lefschetz theorem are outlined. In particular, zeta functions associated to an Axiom A diffeomorphism are compared.


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I thank Magnus Goffeng, Ian Putnam and Robert Yuncken for discussions. In addition, I thank Magnus for encouraging me to publish these results. I also thank the referee for a number of useful suggestions.


  1. 1.
    Bowen, R.: Entropy versus homology for certain diffeomorphism. In: Topology, vol. 13, pp. 61–67. Pergamon, Oxford (1974)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bowen, R., Franks, J.: Homology for zero-dimensional nonwandering sets. Ann. Math. (2) 106(1), 73–92 (1977)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Deeley, R.J., Killough, D.B., Whittaker, M.F.: Dynamical correspondences for Smale spaces. N. Y. J. Math. 22, 943–988 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Deeley, R.J., Killough, D.B., Whittaker, M.F.: Functorial properties of Putnam’s homology theory for Smale spaces. Ergod. Theory Dyn. Syst. 36(5), 1411–1440 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Franks, J.: Homology Theory and Dynamical Systems. CBMS Regional Conference Series in Mathematics, vol. 49, viii+120 pp. American Mathematical Society, Providence, RI (1982)Google Scholar
  6. 6.
    Manning, A.: Axiom A diffeomorphisms have rational zeta functions. Bull. Lond. Math. Soc. 3, 215–220 (1971)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Putnam, I.F.: A homology theory for Smale spaces. Mem. Am. Math. Soc. 232(1094), viii+122 pp. (2014)Google Scholar
  8. 8.
    Ruelle, D.: Thermodynamic Formalism. Encyclopedia of Mathematics and Its Applications, vol. 5, xix+183 pp. Addison-Wesley, Reading, MA (1978)Google Scholar
  9. 9.
    Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)MathSciNetCrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ColoradoBoulderUSA

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