41 problems

A Hamiltonian diffeomorphism is a time-one map of a Hamiltonian flow on a symplectic manifold. For the standard symplectic $2$-torus, a periodic point is a point fixed by some posi…

Zero-entropy conjecture. One has

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 $Y$ be a star-shaped hypersurface in $\mathbb{R}^4$, and call a periodic orbit simple if it is not a multiple cover. Hofer-Wysocki-Zehnder's two-or-infinity conjecture. A Hamil…

Persistence conjecture. For every small $e>0$ there exists $\mu(e)$ for which the orbit $L_1$ has a homoclinic orbit. Moreover, there is a neighborhood $U$ of the curve…

Let the disk carry its standard analytic symplectic structure, and let an analytic symplectic perturbation of the identity mean an analytic symplectic diffeomorphism sufficiently c…

Let $\Sigma\in\mathcal{H}(2n)$ be a compact convex hypersurface, and let $^{\#}\widetilde{\mathcal{J}}(\Sigma)$ denote the number of geometrically distinct closed characteristics o…

Let $\Sigma\in\mathcal{H}(2n)$ be a compact convex hypersurface, and let $^{\#}\widetilde{\mathcal{J}}(\Sigma)$ denote the number of geometrically distinct closed characteristics o…

Let $\Sigma\in\mathcal{H}(2n)$ be a compact convex hypersurface in $\mathbb{R}^{2n}$, where $n\geq2$. A closed characteristic is elliptic when all Floquet multipliers lie on the un…

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

Star-shaped Reeb-orbit conjecture. The corresponding Reeb flow has at least $n$ Reeb orbits.

Let $F$ be a Finsler metric on $S^2$, and call a closed geodesic prime if it is not an iterate of a shorter closed geodesic. Two-or-infinity conjecture for Finsler metrics. Every F…

Let $f$ be a surface symplectomorphism, and interpret “typical” in the conjectural generic or parameterwise sense under discussion. Positive metric entropy conjecture. A typical su…

Let high emergence mean that the number of probability measures needed to describe a dynamical system up to precision $\varepsilon$ grows super-polynomially in $\varepsilon^{-1}$.…

Kolmogorov distinguishes properties typical among Hamiltonian dynamical systems by requiring them to hold for a system and for almost every map in a nearby analytic unfolding…

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 $W$ be a $C^1$ Lagrangian torus invariant under the geodesic flow of a Riemannian metric on $\mathbb{T}^n$, containing $n$ orbits whose canonical projections have well-defined,…

Let $H:T^*\mathbb{T}^n\to\mathbb{R}$ be a Tonelli Hamiltonian, and consider Lagrangian invariant tori at supercritical energy levels. A Hedlund Lagrangian torus is one for which th…

A Hamiltonian dynamical system is said to have a homologically or geometrically unnecessary periodic orbit when it has, for example, a non-contractible or degenerate periodic orbit…

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…

For a Hamiltonian diffeomorphism or homeomorphism $\psi$, $C^0$ rigidity means that there is a sequence of natural numbers $k_i\to\infty$ such that $\psi^{k_i}\to\operatorname{Id}$…

Let $(M,\omega)$ be a symplectically aspherical manifold, meaning that $\omega$ and the first Chern class vanish on every spherical homology class. A Hamiltonian diffeomorphism is…

Mather–MacKay–Meiss–Percival conjecture. The curves $\gamma$ closing the gaps of hyperbolic cantori by arcs of stable or unstable manifold have minimal flux among curves passing th…