2 problems

Let $f:S^2\to S^2$ be an area-preserving diffeomorphism that is Moser generic. Write $\operatorname{Per}(f)$ for the set of periodic points of $f$. Density conjecture. The set of p…

Herman's positive entropy conjecture. For every $\varepsilon>0$ there exists $f\in {\rm Diff}^\infty_\omega(\mathbb D)$ such that $d_\infty(f,id)<\varepsilon$ and