7 problems

Let $f:\mathbb R^2\rightarrow\mathbb R^2$ be a homeomorphism, and suppose that every orbit of $f$ is bounded. Bounded orbit conjecture. Then $f$ has a fixed point. Boyles disproved…

Let $M$ be a connected $n$-manifold. A homeomorphism $f$ of $M$ is almost periodic when the subgroup of $\operatorname{Homeo}(M)$ that it generates has compact closure. Almost peri…

Bounded convex decomposition conjecture. There exists a map

Brown's index-realization conjecture. Every integer $p$ can appear as the index of some iteration in the formula

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 $\mathcal M$ be the Mandelbrot set, and let $\mathcal L_{p/q}$ and $\mathcal L_{p'/q}$ be two limbs associated to a fixed hyperbolic component of $\mathcal M$, with equal denom…

Let $f:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ be an orientation-preserving, fixed-point-free homeomorphism admitting a Lyapunov function…