Problem 103

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How can you write \[ 1 +t  +t^2 \; = \; \Pi _{i=0 }^\infty  (1\pm t^{n_i}) \] in $Z[[ t]] ?$


What Rufus calls the false zeta function is is more commonly called the homology or Lefschetz zeta function.  For this the Lefschetz number of $f^n$ instead of the number of fixed points of $f^n$ is used as coefficient in the formal power series (see [1]).  It is possible that his interest in $1 + t + t^2$ was because it is the homology zeta function of the Plykin attractor.  An expression of the type he seeks would say that there is a periodic point of period $n_i$ which either preserves or reverses the orientation of the unstable manifold according to whether it is $(1- t^{n_i})$ or $(1+ t^{n_i})$.  In [2] it is shown that any formal power series can be expressed essentially uniquely as an infinite product of the type he mentions.  The $n_i$'s and the signs can be solved for recursively, but he may have wanted a closed form expression.


  1. [franks1982homology] Franks JM.  1982.  Homology and dynamical systems.
  2. [franks1985period] Franks J.  1985.  Period doubling and the Lefschetz formula. Transactions of the American Mathematical Society. 287:275–283.

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