4 problems

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…

Two-fixed-points-or-high-periodic-points conjecture. Either $F$ has two fixed points or $F$ has periodic points of arbitrarily high periods.