Problem 104
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…..
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…..
Comments
The infinite product expression for $1/( 1 -t)$ is likely intended as a homology zeta function (see [franks1985period] ). L.-S. Young and J. Franks in [franks1981c2] 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 [MR0431282] .
Cannot decipher…