15 problems

Pugh–Shub conjecture. Stable ergodicity is $mathcal{C}^r$-dense among partially hyperbolic diffeomorphisms.

Let $M$ be a closed orientable $3$-manifold supporting a non-ergodic partially hyperbolic diffeomorphism. A partially hyperbolic diffeomorphism is one whose tangent bundle splits i…

Let $M$ be a compact $3$-manifold, and let ${\cal P}{\cal H}(M)$ be the $C^1$-open set of partially hyperbolic diffeomorphisms of $M$ admitting a dominated splitting … where all th…

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…

In a partially hyperbolic dynamical system, let the central direction (or central bundle) be the invariant direction on which the dynamics may have weak expansion or contraction. N…

Consider volume-preserving partially hyperbolic systems, meaning systems preserving a volume and admitting a partially hyperbolic invariant splitting. Generic exponential mixing co…

A vector field $X$ is called multisingular partially hyperbolic if its chain-recurrence set can be split into finitely many compact invariant sets, each admitting a multisingular p…

A topological Anosov flow is a flow with the topological Anosov properties considered in the paper, and two flows are orbit equivalent when a homeomorphism maps flow orbits onto fl…

Let $f$ be a three-dimensional partially hyperbolic diffeomorphism with invariant splitting $TM=E^s\oplus E^c\oplus E^u$, where each of the stable, center, and unstable bundles…

Let $M$ be a manifold and let $f:M\to M$ be a volume-preserving partially hyperbolic diffeomorphism. Here, “most” is used in the generic sense suggested by the surrounding discussi…

Smooth realization conjecture. Every topological Anosov flow is orbit equivalent to a smooth Anosov flow.

Lower-bound conjecture. The smallest data rate $R_0$ above which stabilization to $S$ can be achieved by a controller receiving state information over a noiseless digital channel s…

Let $M$ be a compact boundaryless manifold, let $\operatorname{Diff}^1(M)$ denote the space of $C^1$ diffeomorphisms of $M$, and let $\mathcal{T}$ be the set of diffeomorphisms exh…

Existence conjecture. There exist nonempty open subsets of $C^1$-stably weakly shadowing symplectomorphisms of dimension at least $6$ such that $M$ is not uniformly hyperbolic for…

Let $L$ be the hyperbolic automorphism under consideration, and let $f$ be homotopic to $L$ and partially hyperbolic in the strongest sense described by the source. Denote the rate…