121 problems

Let $(M,\omega)$ be a compact symplectic manifold, and let a Hamiltonian map on $M$ have more fixed points than are necessarily required by the V. Arnold conjecture. Hofer-Zehnder…

Hunt–Ott and Yuan–Hunt conjectures. For suitable chaotic maps $T$, the periodic optimization property is typical in a probabilistic sense; topologically, it should be typical when…

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…

Density conjecture. The set of parameter values $\beta_2$ at which such special degenerate periodic orbits exist is dense in

Birkhoff's conjecture. The retrograde periodic orbit spans a disk-like global surface of section.

Let $M$ be a compact surface, let $f\colon M\to M$ be a diffeomorphism, let $P_n(f)=\{x\in M:x=f^n(x)\}$, and let $h_{\operatorname{top}}(f)$ denote the topological entropy of $f$.…

Let $T$ be a dynamical system on $W^m$, let $f$ be an observable, and let $\pi(n)$ be the number of periodic trajectories of period $n$. For a trajectory with initial point $w$, wr…

Let $(\mathbb{R}^{2n},\omega_{\mathrm{st}})$ be standard symplectic space, let $H:\mathbb{R}^{2n}\to\mathbb{R}$ be a proper smooth function, and let $\mathcal{AP}_H$ denote the set…

Let a triangle vary through a family of 3-periodic orbits, and consider the Circumellipses in Moses' pencil, namely the Circumellipses whose centers lie on the Feuerbach Circumhype…

Scott's symmetry-line conjecture. Every stable periodic orbit has exactly one or two points on one of these symmetry lines.

Mersenne-period conjecture. For all $n\geq 2$,

Let ${\mathfrak a}0beta\betasubset{\mathfrak a}0\mathbb{R}^2$ be a planar domain, let ${\mathfrak a}0Psi={\mathfrak a}0Psi(\beta)$ denote the interior phase space of its billiard m…

A billiard table is a planar domain in which a billiard ball moves along straight segments and reflects specularly at the boundary; a two-parameter family of periodic orbits is a f…

The dynamics of the outer billiard is defined in the exterior of a convex boundary $\Gamma \in \mathbb{R}^2$. For a point $z_0=(x_0,y_0)\in\mathbb{R}^2$, draw a tangent line to…

Let $H$ be the diagonal split Cartan subgroup in $\operatorname{PGL}_n(\mathbb{R})$ for $n \geq 3$, acting on…

Fix $\rho > 0$. Let $G=\mathbf{G}(\mathbb{R})$ be an $\mathbb{R}$-split real algebraic group, let $\Gamma < G$ be an arithmetic lattice, and let $H$ be a maximal $\mathbb{R}$-split…

Let $n \geq 3$ and let $H$ be the diagonal subgroup acting on $\operatorname{PGL}_n(\mathbb{Z}) \backslash \operatorname{PGL}_n(\mathbb{R})$. Finiteness conjecture for periodic orb…

Let $m$ denote the number of masses on each loop, and let the corresponding periodic orbits be represented by Fourier coefficients $a_k$, normalized by $a_1=1$. Asymptotic shape co…

In a quantum graph, consider a group of isometric periodic orbits and the code words that encode the orbits in that group. Permutation-and-symmetry conjecture. The code words of th…

Let $W\subset\mathbb R^{2n}$ be a star-shaped domain and let $\operatorname{SH}^t(W;\mathbb F)$ denote its filtered symplectic homology with coefficients in a field $\mathbb F$. Th…

Let a Reeb pseudo-rotation be a Reeb flow with finitely many prime closed Reeb orbits. A non-degenerate Reeb flow is one for which all closed Reeb orbits are non-degenerate, and it…

A perturbed family of symmetric interval exchange maps may have periodic orbits that are not symmetric under the system's time-reversal symmetry. Hidden-symmetry conjecture. These…

Iterated Reeb SDM conjecture. The presence of an iterated Reeb SDM implies that the Reeb flow has infinitely many simple Reeb orbits.

Convergence conjecture. The entire orbit $F^n(x_0)$ converges to the period-4 periodic orbit.

Uniform separation conjecture. There exist $\alpha\gg1$ and $T_0\gg1$ such that, for generic obstacles and all $\zeta\in\mathcal G(T)\setminus\tilde{\gamma}$ with $T\geq T_0$, the…