20 problems

Crovisier's conjecture. The bundle $E^c$ has dimension at least two and admits no non-trivial dominated splitting.

Let $f$ be a $C^2$-diffeomorphism and let $K$ be a compact locally maximal invariant set of $f$ admitting a dominated splitting … where $F$ has dimension $1$ and $E^s$ is uniformly…

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 $f$ be a $C^2$ volume-preserving diffeomorphism with an $f$-invariant dominated decomposition $TM=E\oplus F$ such that all Lyapunov exponents along $F$ are positive and all Lya…

Let $r\geq 1$, let $f$ be a $C^r$-diffeomorphism with a basic hyperbolic set $K_f$, and let $\mathcal U$ be a $C^r$-neighborhood of $f$ on which a continuation $g\mapsto K_g$ is de…

Let $T\colon X\to X$ be a basic hyperbolic homeomorphism, for example a full shift, and let $V$ be a smooth vector bundle on $X$. For $r\geq 0$, let $LC^r_{V,T}$ denote the space o…

Let $M$ be a smooth compact manifold and let $f$ be a conservative diffeomorphism. The Oseledets splitting is the invariant splitting supplied by Oseledets' theorem at almost every…

Let $M$ be the compact connected manifold under consideration, let $r>1$, and let $\operatorname{Diff}^r_m(M)$ denote the space of $C^r$ volume-preserving diffeomorphisms of $M$, e…

Herman's conjecture. Under these assumptions, $f$ admits a dominated splitting.

No-splitting conjecture. The class $\Lambda$ admits no non-trivial dominated splitting.

Contracted-bundle conjecture. The bundle $E$ is contracted.

Generic mechanical non-volume-hyperbolicity conjecture. For any integer $r\geq 1$, there is a residual subset $\mathcal R$ of $C^r$-diffeomorphisms $f$ such that if…

Generic mechanical non-domination conjecture. There is a residual subset $\mathcal R$ of $C^r$-diffeomorphisms $f$ such that if $\mathcal C(P,f)$ is not dominated of index $i$, the…

Let $M$ be the connected manifold in the source, let $r>1$, and consider the volume-preserving diffeomorphism space $\operatorname{Diff}^r_m(M)$ with its $C^1$ topology. A diffeomo…

Let $\Lambda$ be an aperiodic class for a $C^1$-generic diffeomorphism $f$, and let … be a dominated splitting such that $E^s$ and $E^u$ are respectively the maximal uniformly cont…

Let $M$ be a compact manifold, let $\omega$ be a volume form, and let $\operatorname{Diff}^r_\omega(M)$ denote the space of $C^r$ volume-preserving diffeomorphisms, with $r>1$. A d…

Let $M$ be the compact manifold equipped with volume measure $m$, and let $\mathrm{Diff}_m^1(M)$ denote the space of $C^1$ volume-preserving diffeomorphisms. A diffeomorphism is no…

Bonatti's tameness conjecture. The set ${\cal O}_0(M)$ consists of tame diffeomorphisms. The source notes that ${\cal O}_0(M)$ is nonempty and that it is an open question whether i…

Let $\Lambda$ be the singular attractor containing the hyperbolic singularities $s_0$ and $s_1$ of different indexes. On the part of $\Lambda$ away from the equilibria, suppose a d…