4 problems

A contact form $\lambda$ is said to satisfy the strong closing property if, for every non-zero function $g\geq 0$ or $g\leq 0$, there exists $s\in[0,1]$ such that $(1+sg)\lambda$ h…

Let $M$ be a compact manifold equipped with a symplectic form $\omega$, and let $\operatorname{Diff}^{r}(M,\omega)$ be the space of $C^{r}$ diffeomorphisms preserving $\omega$, whe…

Consider a dynamical system and a nonwandering point, meaning a point which does not have a neighborhood disjoint from its images under iterates. Poincaré's closing lemma conjectur…

Closing conjecture. There exist smooth metrics arbitrarily close to $g$ in the $C^2$ topology such that the unit-speed geodesic starting at $x$ with initial velocity $v$ is periodi…