5 problems

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

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…

Avila–Bochi conjecture. For a generic conservative diffeomorphism $f\in \operatorname{Diff}^1_m(M)$, one of the following alternatives holds: all Lyapunov exponents of $f$ are zero…

Theorem 1 concerns a compact orientable surface $S$ with a smooth volume form and asserts that a residual subset of pairs in…