11 problems

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 $F$ 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 $Y$ 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…

Let $(M,\omega)$ be a symplectic manifold and let $H:M\to\mathbb{R}$ be a Hamiltonian. An energy level is a level set $H^{-1}(c)$, and a hypersurface $Y\subset M$ is of contact type…

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

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 $(S,\Omega)$ be an analytic closed symplectic surface, and let $f\in \operatorname{Diff}^{\omega}_{\Omega}(S)$ be an analytic symplectic diffeomorphism displaying an elliptic p…

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…

Local $C^\infty$-closing property. The standard symplectic vector space $({\mathbb R}^{2n},\Omega_{\textrm{standard}})$ has the $C^\infty$-closing property for every $n\geq 2$. In…

Let $\Delta_0$ and $\Delta_1$ denote the instability quantities associated with an area-preserving twist diffeomorphism, and let $V$ be a $C^2$ potential. Instability-gap conjectur…

Generic symplectic Oseledets splitting conjecture. For a generic $f\in \operatorname{Diff}^1_\omega(M)$ either all Lyapunov exponents vanish almost everywhere, or else the Oseledet…