15 problems

Let $\alpha\colon\mathbb{R}^k\curvearrowright M$ be a higher rank Anosov action, meaning an Anosov action with $k\geq 2$, and suppose that it has no rank $1$ factors. Katok–Spatzie…

Kalinin–Spatzier conjecture. Every Anosov action of $\mathbb{R}^k$, for $k\geq 2$, is algebraic.

Let an action of $\mathbb Z^k$ or $\mathbb R^k$, with $k\geq 2$, act by commuting diffeomorphisms of a compact manifold and contain at least one Anosov element. Call the action irr…

An Anosov action of $\mathbb Z^k$ or $\mathbb R^k$ is an action containing an element that is normally hyperbolic relative to the orbit foliation. An action is irreducible in the s…

Classification conjecture. Every irreducible higher-rank ${\mathbb Z}^k$- or ${\mathbb R}^k$-Anosov action on any compact manifold is smoothly conjugate to an algebraic action.

Algebraicity conjecture. Invariant ergodic measures for Anosov actions without rank-one factors are always algebraic.

Global smooth classification conjecture. All Anosov actions of such lattices are smoothly equivalent to one of the above algebraic actions.

Let an algebraic abelian Anosov action have rank $\kappa\geq 2$, and assume the rigidity conjecture so that its invariant-measure classification reduces to homogeneous dynamics. Th…

Katok–Katok cohomological conjecture. For such standard actions, one has, for $j=1,\dots,\kappa-1$,

Higher-rank rigidity conjecture. Higher-rank $\mathbb{R}^\kappa$-Anosov actions are always smoothly conjugate to several models, mostly of algebraic nature.

Let $G$ be an abelian Lie group of dimension $k$. A codimension-one faithful Anosov action is an Anosov action of $G$ whose unstable direction has dimension one and which is faithf…

Infranilmanifold global rigidity conjecture. All irreducible Anosov genuine $\mathbb{Z}^r$-actions on compact infranilmanifolds should be $C^\infty$-conjugate to actions by affine…

Let $\phi$ be an irreducible codimension-one Anosov action of $\mathbb R^k$ on a manifold of dimension at least $k+3$, with $k\geq2$. A smoothly conjugate action is one related by…

Smooth classification conjecture. All irreducible Anosov actions of $\mathbb{Z}^k$ or $\mathbb{R}^k$ are smoothly conjugate to algebraic actions.

Let an irreducible higher-rank ${\mathbb Z}^d$ Anosov action be a smooth action of ${\mathbb Z}^d$ with the stated irreducibility and Anosov properties. Spatzier's conjecture. Any…