8 problems

Let $M$ be a closed manifold, let $f:M\to M$ be a diffeomorphism preserving a volume form $\mu$, and let $\mathrm{Diff}^r_\mu(M)$ denote the corresponding space of $C^r$ volume-pre…

Let $S$ be a closed surface, and let $k$-tuples of volume-preserving diffeomorphisms be equipped with the relevant genericity notion. Dolgopyat–Krikorian conjecture. Stable ergodic…

Let $(M,g)$ be a closed Riemannian manifold and let $\omega$ be its volume form. For an integer $r$, let $\Lambda^{r}_{\omega}(M)$ denote the space of $C^1$ expanding $r$-folds of…

Let $X$ be a conservative $C^1$ vector field and let $\Gamma$ be a singular hyperbolic set for $X$. A singular adapted metric is a Riemannian metric satisfying the contraction, dom…

Let $M$ be a compact manifold and let $r\in\mathbb N\cup\{\infty\}$. An iterated function system consists of a finite family of homeomorphisms of $M$; it is transitive if the semig…

Let $M$ be a compact Riemannian manifold, let $r$ be sufficiently large, and let $m\geq 2$ be an integer. An $m$-tuple of $C^r$ volume-preserving diffeomorphisms of $M$ is called…

Escaping-trajectories conjecture. There are real-analytic divergence-free vector fields in the closed unit ball with an open and dense subset filled with escaping trajectories, and…

Let $\Gamma$ be a group that is generic in an appropriate model for random groups, let $M$ be a compact manifold, and consider an action of $\Gamma$ on $M$. A smooth volume form is…