5 problems

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 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 $M$ 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…

The non-trivial self-cover conjecture. The infra-nilmanifold $\Gamma \backslash G$ admits a non-trivial self-cover if and only if the nilmanifold $N \backslash G$ admits an expandi…

The expanding-map lifting conjecture. The infra-nilmanifold $\Gamma \backslash G$ admits an expanding map if and only if the nilmanifold $N \backslash G$ admits an expanding map.