62 problems

Let $f$ be a transitive partially hyperbolic diffeomorphism of a $3$-manifold. A diffeomorphism is a variable time map of a flow if the time spent along each orbit is given by a su…

Let $f$ be a partially hyperbolic diffeomorphism on a 3-manifold, with invariant splitting $TM=E^s\oplus E^c\oplus E^u$. Write $E^{cu}=E^c\oplus E^u$ and $E^{sc}=E^s\oplus E^c$. Th…

Let $f$ be a volume-preserving partially hyperbolic diffeomorphism of a $3$-manifold with invariant splitting $TM=E^s\oplus E^c\oplus E^u$. A torus is tangent to the…

A diffeomorphism $f:M\to M$ is dynamically coherent partially hyperbolic when its partially hyperbolic structure admits invariant center foliations. It is plaque expansive if there…

Physical-measure conjecture. The attractors constructed in Section admit a unique physical/SRB measure with a full volume ergodic basin.

Let $A$ be a hyperbolic toral automorphism and let $f$ be a volume-preserving, possibly non-ergodic, $C^1$-small perturbation of $T_A\times\operatorname{id}$. Write…

A volume-preserving partially hyperbolic diffeomorphism is a diffeomorphism preserving a volume measure and admitting a partially hyperbolic invariant splitting; when its center fo…

ASE-to-NU2SE conjecture. A partially hyperbolic compact invariant set for a smooth flow satisfying the ASE condition (pASE) also satisfies the NU2SE (NUpSE) condition.

Let $f$ be a C^ diffeomorphism, let $\widehat{\mu}=\int \mu_\varpi\,d\widehat{\mathbf{p}}(\varpi)$ be an ergodic generalized $u$-Gibbs measure, and let $\varpi$ denote the disks…

Let $f: M^3\rightarrow M^3$ be a $C^r$ ($r > 1$) conservative partially hyperbolic diffeomorphism of a closed 3-manifold that fails to be ergodic. Here, $E^s$ and $E^u$ denote the…

Let $f$ be a $C^r$ ($r>1$) conservative partially hyperbolic diffeomorphism on a 3-manifold, with invariant bundles $E^s$ and $E^u$. Strong Ergodicity Conjecture. If $f$ is non-erg…

Let $M^3$ be an orientable closed 3-manifold and let $f:M^3\to M^3$ be a $C^r$ ($r>1$) non-ergodic partially hyperbolic diffeomorphism. Weak Ergodicity Conjecture. Then $M^3$ must…

A partially hyperbolic diffeomorphism is a diffeomorphism with an invariant splitting into stable, center, and unstable bundles, with the stable and unstable directions dominating…

Let $M$ be a Kähler manifold and let $f$ be a holomorphic partially hyperbolic diffeomorphism of $M$, with center distribution $E^c$. Kähler center-holomorphicity conjecture. The c…

Let $f$ be a holomorphic partially hyperbolic diffeomorphism, with center distribution $E^c$. Real-analyticity conjecture. The center distribution $E^c$ is real analytic. The conje…

Let $f$ be a holomorphic partially hyperbolic diffeomorphism on a complex $3$-manifold. Three-dimensional classification conjecture. Up to passing to a finite cover, $f$ is holomor…

Topological Anosov conjecture. Any such $f$ must be topologically Anosov.

Let $M$ be a compact manifold and let $f\in\operatorname{PH}^r(M)$ be a dynamically coherent partially hyperbolic diffeomorphism. A bi-infinite sequence $\{x_i\}_{i\in\mathbb{Z}}$…

Let $r\in[2,+\infty]$, and consider $C^{r}$ partially hyperbolic diffeomorphisms, whether volume preserving or not. A diffeomorphism is stably accessible if it has the accessibilit…

Let $f$ be a $C^{2}$ partially hyperbolic, volume-preserving diffeomorphism, and suppose that $f$ has the essential accessibility property, meaning that every measurable set satura…

Let $M^{3}$ be a three-dimensional smooth compact manifold, let $m$ be a smooth volume measure, and let $\operatorname{Diff}^{1}_{m}(M^{3})$ denote the volume-preserving $C^{1}$ di…

Let $f$ be a perturbation in the setting of the paper, and suppose conclusion (1) of the centralizer classification applies, so that its relevant smooth centralizer is virtually…

Let $T_A$ be a general hyperbolic toral automorphism, let $f$ be the perturbation under consideration, and let $\mathcal{Z}_{\infty}(f)$ denote its smooth centralizer. For…

Gogolev–Maimon–Kolgomorov conjecture. For all analytic diffeomorphisms $f$ in a sufficiently small neighborhood of $A$, the strong unstable foliation ${\mathcal W}^{uu}$ is transit…

Let $A\colon \mathbb T^3\to\mathbb T^3$ be the hyperbolic automorphism induced by the displayed integral matrix, and let $f$ be an analytic diffeomorphism in a sufficiently small n…