36 problems

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

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

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 $M$ be a closed Riemannian manifold, and let $f:M\to M$ be a partially hyperbolic diffeomorphism, meaning that $TM$ admits a $Df$-invariant splitting … where $E^s$ is uniformly…

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…

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…

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 $f$ be a conservative partially hyperbolic diffeomorphism of a 3-manifold. Hertz–Hertz–Ures ergodic conjecture. If $f$ is non-ergodic, then there is a 2-torus tangential to…

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…

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…

Let $A\colon \mathbb T^3\to\mathbb T^3$ be the hyperbolic automorphism induced by the displayed integral matrix, and let $W^{uu}_f$ denote the one-dimensional strong unstable folia…

Let $M$ be a manifold and let $f:M\to M$ be a partially hyperbolic diffeomorphism. For $x\in M$, the accessibility class $AC(x)$ is the set of points that can be joined to $x$ by a…

Let $M$ be a three-dimensional manifold and let $f$ be a partially hyperbolic diffeomorphism of $M$, with stable, center, and unstable bundles. Three-dimensional continuity conject…

Let $M$ be a compact manifold, and let $f$ be a $C^1$ partially hyperbolic diffeomorphism of $M$ with a decomposition $TM=E^s\oplus E^c\oplus E^u$ into invariant continuous sub-bun…

Let $M^3$ be a 3-dimensional manifold and let $f:M^3\rightarrow M^3$ be a partially hyperbolic diffeomorphism. A $cu$-torus is a 2-torus tangent to the center-unstable bundle…

Let $f$ be a partially hyperbolic diffeomorphism of a 3-manifold. Pujals' classification conjecture. If $f$ is transitive, then $f$ is finitely covered by one of the following: a p…

Consider orientable 3-manifolds supporting partially hyperbolic dynamics. Hertz–Hertz–Ures weak dynamical coherence conjecture. The only orientable 3-manifolds supporting non-dynam…

Consider an orientable 3-manifold and a conservative partially hyperbolic diffeomorphism on it. Hertz–Hertz–Ures weak ergodicity conjecture. The only orientable 3-manifolds admitti…

Let $X$ be a simply connected closed manifold or a simply connected finite simplicial complex, and let $p\colon E\to X$ be a fiber bundle with closed-manifold fiber $M$. Assume tha…