9 problems

Anosov-flow conjecture. Every robustly transitive action $\varphi\in A^2(\mathbb{R}^2,N)$ with a dense orbit is defined by an Anosov flow, without assuming that the dense orbit is…

Let $S^3$ denote the three-dimensional sphere. A diffeomorphism is robustly transitive if its transitivity persists under sufficiently small perturbations in the $C^1$ topology. Sp…

A diffeomorphism is partially hyperbolic if its tangent bundle admits a splitting … where $E^s$ is uniformly contracting and $E^u$ is uniformly expanding. Partial hyperbolicity con…

Let $X$ be a robustly transitive contact flow, meaning a contact flow that remains topologically transitive under the relevant perturbations within the class of contact flows. Cont…

Let $g$ be a Riemannian metric whose geodesic flow is robustly transitive, meaning that $g$ admits a $C^2$-neighbourhood such that every metric in that neighbourhood has topologica…

Let $M$ and $N$ be closed manifolds of arbitrary finite dimension. Suppose that $M$ supports an expanding map $F$. Product-map conjecture. The product $M\times N$ supports a $C^1$…

Unstable-cone expansion conjecture. Every robustly transitive endomorphism in dimension two has unstable cones, and every curve whose tangent vector at each point lies in the corre…

Let $M$ be a symplectic manifold and let $r$ be the differentiability class used for the $C^r$ topology. A symplectic diffeomorphism of $M$ is a diffeomorphism preserving the sympl…

Let $M$ be a compact manifold, and consider $C^r$-generic partially hyperbolic symplectic or conservative diffeomorphisms of $M$. A diffeomorphism is robustly transitive if transit…