11 problems

Let $f$ be a homeomorphism of the two-dimensional torus homotopic to the identity, and let $F$ be a lift of $f$. Its rotation set is … A rotation set is thus a compact convex subse…

Let the full rotation set and the admissible rotation set be the rotation sets associated with the billiard system with one obstacle, as described in the paper. The admissible rota…

Bounded convex decomposition conjecture. There exists a map

Let $\rho$ be a parameter, let $\Lambda_\rho$ be the associated set, and let $R_\rho(\Lambda_\rho)$ denote its roundness. For each $k\in\mathbb{N}\setminus\{0,1\}$, consider the pa…

Diagonal monotonicity conjecture. If $0\leq\alpha\leq\tilde\alpha$, then

A homeomorphism $fbrcolon torus^2to torus^2$ is periodic point-free if it has no periodic points, and it is a pseudo-rotation when its rotation set consists of a single vector. Per…

Let $f$ be a homeomorphism of ${\bf T}^2$ isotopic to the identity, preserving a full-support probability measure $\mu$, and let $\check f$ be a lift of $f$ to ${\bf R}^2$. Assume…

Let $f\in{\mathrm{Homeo}}_0(\mathbb{T}^2)$ have a rotation set $\rho(F)$ with non-empty interior, and let $C\subseteq{\mathrm{int}}(\rho(F))$ be a continuum. A realization conjectu…

Let $f\in\operatorname{Homeo}_0(\mathbb{T}^2)$, and let $\rho(f)\subset\mathbb{R}^2$ be its rotation set. Suppose that $\rho(f)$ is a nondegenerate line segment. Franks–Misiurewicz…

Uniform horizontal bounded-deviation conjecture. The displacement $\widehat{f}^n(z)-z$ should be uniformly bounded in the horizontal direction.

Let $f\colon \mathbb{T}^2\to\mathbb{T}^2$ be a homeomorphism homotopic to the identity, let $\widehat{f}$ be a lift of $f$ to $\mathbb{R}^2$, and let $\rho(\widehat{f})$ denote its…