17 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…

Albach's conjecture. Either the flow has exactly two periodic orbits, or

Let $M$ be a manifold of dimension at least $2$. Let $(S^{\ast}M,\xi)$ be its spherical cotangent bundle with the standard contact structure, let $\alpha$ be a contact form with…

Let $(\xi^+,\xi^-)$ be a bicontact structure supporting an Anosov flow $X$. A Reeb flow is bitransverse when it has the bitransversality property required in the paper, and a flow…

Let $(xi^+,xi^-)$ be a bicontact structure supporting an Anosov flow $X$. The bicontact trichotomy conjecture. Exactly one of the following holds: 1. $xi^+$ is Anosov contact, it a…

Let $x$ be an isolated closed Reeb orbit, let $k$ be a positive integer, and let $\operatorname{CH}(x;{\mathbb Q})$ denote its local contact homology. Let…

A pseudo-rotation is a Hamiltonian system, Hamiltonian diffeomorphism, or Reeb flow with finitely many periodic orbits. Anosov–Katok realization conjecture. All pseudo-rotations ar…

Let $M$ be the smooth boundary of a star-shaped domain in ${\mathbb R}^{2n}$. A Reeb pseudo-rotation is a Reeb flow on $M$ with finitely many prime closed orbits. HZ-conjecture. Th…

Let the sequential barcode entropy and total persistence entropy be the two Reeb-flow variants of barcode entropy described in the source, and let Theorem A and Corollary C denote…

Let $(M,\alpha)$ be a contact manifold, let $X_\alpha$ be its associated Reeb vector field, and let $\phi$ be an almost complex structure adapted to $(M,\alpha)$. The Reeb flow is…

Reeb-flow entropy stability conjecture. The functional $h_{\rm top}$ is lower semi-continuous with respect to $d_{C^0}$.

Let $\Sigma$ be a closed energy surface equipped with a Reeb flow, that is, the flow of the Reeb vector field associated to a contact form on $\Sigma$. Rabinowitz–Weinstein conject…

Let $(S^{2n+1},\xi_{std})$ be the standard contact sphere, and let $\alpha$ be any contact form on it. A closed orbit is simple if it is not an iterate of another closed orbit. Clo…

Let $(M,\xi)$ be a contact 3-manifold. Let $X_\alpha$ be a Reeb vector field and $J$ a complex structure on the contact distribution, and let $\theta'$ denote the instantaneous rot…

Let $\alpha$ be a contact form on $S^3$ obtained as the lift of a contact form on $SO(3)$ induced by a Finsler or Riemannian geodesic flow, and let $T_{\min}(\alpha)$ and…

Let $\Sigma$ be a convex hypersurface in $\mathbb{R}^{2n}$, equipped with the Reeb flow induced by its contact form. A periodic orbit is elliptic if every eigenvalue of its lineari…

Positive entropy conjecture. If $M$ is fast, then every Reeb flow on $(S^*M,\xi)$ has positive topological entropy.