46 problems

A closed manifold is infra-nilmanifold if it is homeomorphic to a compact quotient of a nilpotent Lie group by a discrete group of isometries. An Anosov diffeomorphism is a diffeom…

Classification conjecture. Any Anosov diffeomorphism is topologically conjugate to an Anosov automorphism of an infranilmanifold.

Mendes's conjecture. The diffeomorphism $f$ is topologically conjugate either to a translation or to a hyperbolic linear map.

A diffeomorphism $f:M\to M$ of a compact Riemannian manifold is Anosov if $TM$ admits a $Df$-invariant splitting $E^s\oplus E^u$ with uniform contraction on $E^s$ and uniform expan…

Continuity conjecture. The Lyapunov exponents vary continuously as functions of $f$ in the space of $\mathcal{C}^2$ volume-preserving Anosov diffeomorphisms on $\mathbb T^3$.

Let $M$ be a smooth manifold, let $f\in\operatorname{Diff}^\infty(M)$ be an Anosov diffeomorphism, and let $\mathcal{J}_M(f)$ denote the set of complex structures on $M$ preserved…

Let $M$ be a connected manifold and let $f\colon M\to M$ be an Anosov diffeomorphism. The diffeomorphism $f$ is topologically transitive if it has a dense orbit in $M$. Transitivit…

Anosov diffeomorphism classification conjecture. Every Anosov diffeomorphism is topologically conjugate to a hyperbolic infra-nilmanifold automorphism.

Katok's conjecture. If $g$ is Hölder conjugated to an Anosov diffeomorphism $f$, then $g$ is also an Anosov diffeomorphism.

Algebraicity conjecture. Anosov diffeomorphisms are always of algebraic nature, up to topological conjugacy.

Let $M$ be a compact differentiable manifold and let $f\colon M\to M$ be an Anosov diffeomorphism, meaning that the tangent bundle admits a continuous invariant splitting … with…

Let $M$ be a smooth manifold and let $f:M\rightarrow M$ be an Anosov diffeomorphism. An algebraic Anosov diffeomorphism is an Anosov diffeomorphism induced by an automorphism of a…

Let $M$ be a compact manifold with an Anosov diffeomorphism whose stable and unstable foliations are smooth. Denote the reduced leafwise cohomologies associated with these foliatio…

Franks's conjecture. The diffeomorphism $f$ is $C^0$-conjugate to a hyperbolic infranilautomorphism.

Franks–Manning conjecture. Smooth Anosov diffeomorphisms can exist only on infranilmanifolds.

Let $f:\mathbb T^N\to\mathbb T^N$ be a $C^{s}$, $s>1$ Anosov diffeomorphism with linearization $A$, and suppose that $A$ is irreducible. Say that $f$ and $A$ have the same periodic…

Let $M$ be a smooth manifold, let $f\in\operatorname{Diff}^\infty(M)$ be an Anosov diffeomorphism, and let $\mathcal Z^\infty(f)$ denote its $C^\infty$-centralizer. Katok--Spatzier…

Let $M$ be a compact manifold and let $f\colon M\to M$ be an Anosov diffeomorphism. Anosov--Smale conjecture. Every Anosov diffeomorphism is topologically conjugate to an affine au…

Let $f:M\to M$ be a $C^k$-smooth Anosov diffeomorphism. Denote its stable and unstable foliations by $\mathcal{F}^s_f$ and $\mathcal{F}^u_f$, respectively. Flaminio–Katok conjectur…

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

An Anosov diffeomorphism on a closed manifold $M$ is a diffeomorphism $f:M\to M$ preserving a continuous splitting $TM=E^s\oplus E^u$ such that $df$ exponentially contracts vectors…

Let $M$ be a compact manifold and let $f\colon M\to M$ be an Anosov diffeomorphism, with invariant stable and unstable distributions $E^s$ and $E^u$. A diffeomorphism is finitely c…

Let $f$ be an analytic Anosov diffeomorphism of a compact surface, and let $H$ be an analytic function on the surface whose zero level set has no critical points. Conditional measu…

Foliation-regularity conjecture. The conjugacy between $f$ and $A$ should have the same $C^2$ regularity.

Rational-form invariance conjecture. If $\mathfrak{n}_{\mathcal{G}}^\mathbb{Q}$ admits an Anosov automorphism, then every rational form $\mathfrak{m}^\mathbb{Q}$ of…