29 problems

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

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 $H_0$ be the proper Hamiltonian on ${\mathbb R}^{2n}$ under discussion, let $K\subset(0,\infty)$ be a compact set of measure zero, and let…

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 $Y$ 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…

Density of periodic orbits. Every topological Anosov flow $\theta\in\mathcal{TA}\mathcal{F}(M)$ possesses a dense set of periodic orbits.

Let $F$ be a $C^k$ vector field on $S^3$, with $k\geq 3$. Let $\Lambda$ be a compact connected invariant set and $S$ a cross-section for $\Lambda$ such that the first-return map…

Let $F$ be a smooth vector field on $S^3$ with fixed points $\{x_1,\ldots,x_n\}$, and let $H$ be a one-dimensional flow-invariant set such that each component $C$ of…

Ordering conjecture.

Shelukhin's Hofer-Zehnder conjecture. If

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…

Periodic-cylinder conjecture. For $k=q+1$ and $q$ a positive integer, the generalized Chazy differential system has an invariant cylinder foliated by periodic orbits.

Periodic-orbit conjecture. For a no-slip billiard with an external force, $2$-periodic non-path-reversing orbits exist between any two accessible points on an arbitrary table, and…

Generic Conley conjecture. On every closed symplectic manifold, almost all Hamiltonian diffeomorphisms have infinitely many simple contractible periodic orbits.

Marchal's conjecture. The three-body equilateral triangle of Lagrange and the three-body figure eight are in the same continuation class.

Let $f:[0,1]\to[0,1]$ be continuous, let $x\in[0,1]$, and let $s\alpha(x)$ be the special $\alpha$-limit set of $x$. Period-three coexistence conjecture. If $s\alpha(x)$ contains a…

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…

Tresser's conjecture. Generically, maps in the boundary of positive topological entropy have a set of periodic orbits which, except for a finite subset, consists of infinitely many…

Let $(M,\omega)$ be a $b^m$-symplectic manifold, and let $H\in C^\infty(M)$ be a proper smooth Hamiltonian. A singular periodic orbit is a periodic orbit in the singular Hamiltonia…

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…

Assume that $\mathcal{K}$ is not central. Let $M einmathbb{N}$ be as in the symmetry condition, and let $(Deltatheta){\min}$ be the minimum total angle compatible with $\mathcal{K}$.…

Local hyperbolization conjecture. There exists a function $\tilde f\in\mathcal{N}$ such that $p$ is a hyperbolic periodic solution of the equation with nonlinearity $\tilde f$.

Typically periodic optimization (TPO) conjecture. The set $V_{Per}$ contains an open dense subset of $V$.