Problem 104

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 Note: \[ \frac {1}{1-t} \; = \; \Pi _{n= 0}^\infty (1 +t^{2^n}) \] in $Z[[ t]]$. Is there a $C^2$ map $\mathbb D^2 \to \mathbb D^2$ so that.....


Cannot decipher...

The infinite product expression for $1/( 1 -t)$ is likely intended as a homology zeta function (see [1]).  L.-S. Young and J. Franks in [2] construct a $C^1$ diffeo of $D^2$ with one periodic point of period $2^n$ for each $n$.  They are all saddles and there are no other periodic points.  The homology zeta function for this example is$1/(1-t)$.  Possibly Rufus was asking if this could be done $C^2$. The case when the diffeo is $C^1$ was done in  [3].


  1. [franks1985period] Franks J.  1985.  Period doubling and the Lefschetz formula. Transactions of the American Mathematical Society. 287:275–283.
  2. [franks1981c2] Franks J, Young L-S.  1981.  A C2 Kupka-Smale diffeomorphism of the disk with no sources or sinks. Dynamical Systems and Turbulence, Warwick 1980. :90–98.
  3. [MR0431282] Bowen R, Franks J.  1976.  The periodic points of maps of the disk and the interval. Topology. 15:337–342.

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