11 problems

Let $L$ be an irreducible hyperbolic toral automorphism, and let $f$ be a C^ diffeomorphism of the torus that is $C^1$-conjugate to $L$. Gogolev's smooth rigidity conjecture. The…

Let $A$ be a hyperbolic toral automorphism and let $f$ be a volume-preserving, possibly non-ergodic, $C^1$-small perturbation of $T_A\times\operatorname{id}$. Write…

Let $T,S\in\operatorname{SL}(d,\mathbb{Z})$, and let $\mathcal{D}$ be a Borel subset of $\mathbb{T}^d$ of positive measure. Boshernitzan's conjecture. … This is a measurable analog…

Almost one-to-one conjecture. The restriction $\xi|_{V_\beta}$ is almost one-to-one.

Let $T$ be a Pisot automorphism of the torus, algebraically conjugate to the companion toral automorphism $T_{\beta}$ of a Pisot unit $b2$, so that there exists…

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 $L$ be a hyperbolic automorphism of $d53b^d$, and let $f$ be a $C^r$ perturbation of $L$ that is conjugate to $L$ by a conjugacy $H$ as in Theorem 1.1. Gogolev's conjecture. Th…

Let $T_A$ be a general hyperbolic toral automorphism, let $f$ be the perturbation under consideration, and let $\mathcal{Z}_{\infty}(f)$ denote its smooth centralizer. For…

Let $f$ be an irreducible, nonhyperbolic polynomial, and let $\bar{Z}_f$ be the shift space constructed in Theorem 2. A shift space $\Omega\subset\mathsf{A}^{\mathbb{Z}}$ with fini…

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…

Drift conjecture. The drift vector is