13 problems

Let $M$ be a compact complex manifold and let $f:M\to M$ be a holomorphic Anosov diffeomorphism. A complex infra-nilmanifold is a quotient of a complex nilmanifold by a finite grou…

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…

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…

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 closed manifold and let $f:M\to M$ be an Anosov diffeomorphism. A hyperbolic affine infra-nil-automorphism is an affine automorphism of an infra-nil-manifold whose lin…

Let $M$ be a closed manifold and let $f\colon M\to M$ be an Anosov diffeomorphism, meaning a diffeomorphism with a hyperbolic splitting of the tangent bundle. A hyperbolic infra-ni…

Let $f$ be a $C^2$ volume-preserving Anosov diffeomorphism of $\mathbb{T}^2$, and let $\Lambda:\operatorname{Diff}^2_m(\mathbb{T}^2)\to\mathbb{R}$ be the functional defined in the…

The conjecture from the 1960s. If there is an Anosov diffeomorphism $f$ on $M$, then some finite cover of $M$ is a nilmanifold. Furthermore, lifting $f$ to the finite cover yields…

Let $M$ be the compact manifold under consideration, let $f\in\operatorname{Diff}^1(M)$, and let $\mathcal{TLS}$ denote the set of diffeomorphisms with the two-sided limit shadowin…

Let $\mathcal U$ be the neighborhood of the hyperbolic toral automorphism considered in the source. Local counterexample conjecture. For any $f\in\mathcal U$ there exists…

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…

Let $L$ be a hyperbolic automorphism of $\mathbb T^d$, $d\geq 3$, with simple real spectrum and irreducible characteristic polynomial over $\mathbb Z$. Let $\mathcal U$ be the neig…

Let $f$ and $g$ be topologically conjugate $C^r$, $r>1$, Anosov diffeomorphisms of $\mathbb T^3$ with coinciding periodic data, meaning that the differentials of corresponding retu…