4 problems

Herman's conjecture. Under these assumptions, $f$ admits a dominated splitting.

Pesin's density conjecture. Arbitrarily close to $f$ in $\operatorname{Diff}^r_m(M)$, there is a diffeomorphism $g\in \operatorname{Diff}^r_m(M)$ without zero Lyapunov exponents on…

Let $M$ be a compact manifold, let $\omega$ be a volume form, and let $\operatorname{Diff}^r_\omega(M)$ denote the space of $C^r$ volume-preserving diffeomorphisms, with $r>1$. A d…

Let $M$ be a compact surface, and consider the space of $C^1$ area-preserving maps of $M$ with its $C^1$ topology. A map is Anosov if it has a uniformly hyperbolic invariant splitt…