62 problems

Arnold's conjecture for non-degenerate Hamiltonians.

Arnold's conjecture. The number of fixed points of $f$ satisfies

Let $x_1(t),\ldots,x_N(t)$ be the trajectories of the Hamiltonian flow generated by $E^{(N)}$, and let $\delta_N((x_1(t),\ldots,x_N(t)))$ be their empirical measure. Let $\rho_t$ a…

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…

Let $H$ be the Kepler–Heisenberg Hamiltonian, and let $z(t)$ denote the vertical coordinate of a zero-energy orbit. An orbit avoids the $z$-axis if it never meets the set where the…

Consider a near-integrable Hamiltonian system with slow action variables $y$ and perturbations of the form … where $x\in\mathbb{T}^n$, $y\in\mathbb{R}^n$, $t\in\mathbb{T}$, and…

Arnold–Givental conjecture. Assume that $(M,\omega,\rho)$ is a closed real symplectic manifold and $\phi_H$ is a Hamiltonian diffeomorphism such that $L$ and $\phi_H(L)$ intersect…

Extension criterion for first class functions. A function $f$ on the constraint set $\mathscr{C}$ extends to a first class function if and only if it is constant along all the inte…

Let $(n_0,j_0)$ be a pair of natural numbers with $n_0\geq2$, $j_0\geq1$, and set … Suppose that the only natural number solution of the Diophantine equation … is $(j_0,n_0)$. Peri…

The spherical capillary water waves system is considered on the sphere, with small-amplitude time-periodic solutions as the objects of interest. Periodic-solution conjecture. There…

Let $X$ be a symplectic cluster $4$-manifold equipped with a cluster presentation $\{\pi,L_1,\dots,L_n\}$. A fiber means a fiber of $\pi$ over a point of the base. Non-displaceabil…

Let $(M,\omega)$ be a symplectic manifold, let ${\rm Diff}_{\rm Ham}(M)$ denote the Hamiltonian diffeomorphism group, and let ${\rm Diff}_{\rm Ham}^k(M)$ be the subspace of diffeom…

Let $M$ be a manifold, let $\sigma$ be a symplectic form on $M$, and let $H$ be the twisted geodesic Hamiltonian on $T^*M$. Magnus conjecture. For every $M$ and any symplectic form…

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 $(M,\omega)$ be a compact symplectic manifold, and let $\gamma$ be the Floer-theoretic pseudometric on the universal cover of $\mathrm{Ham}(M)$. Assume that $\omega$ vanishes o…

Diffusion conjecture. For every $\varepsilon>0$, there exists a law-defining datum $\mathcal{D}$ such that the push-forward measure $(ev_p)_*\mu_{\operatorname{Ham}}^{\mathcal{D}}$…

Let $M$ be a compact or Liouville symplectic manifold, and let $L\subseteq M$ be a compact, relatively exact Lagrangian. Let $\tilde\phi:L\to L$ be a diffeomorphism extending to a…

Let $(W,\omega)$ be a Liouville domain, and let ${\rm SH}^*(W,\omega)$ denote its symplectic homology. Equip $\operatorname{Ham}(W,\omega)$ with the metric $d_{\gamma}$. Infinite q…

Let $(M,\omega)$ be a compact symplectic manifold, and let $H:\mathbb{R}\times M \to \mathbb{R}$ be a time-dependent Hamiltonian function. Suppose that the solutions of period $1$…

A transition chain is a chain of hyperbolic invariant tori whose stable and unstable manifolds can be shadowed by drifting orbits in a near-integrable Hamiltonian system. Arnold's…

Let $(M,\omega)$ be a closed connected symplectic manifold. Denote by $\operatorname{Symp}(M,\omega)$ the group of symplectomorphisms, by $\operatorname{Symp}_0(M,\omega)$ its iden…

Nonlinear Perron–Frobenius conjecture. There exists a value $E^\star(\boldsymbol{\gamma})$ such that: for $E>E^\star(\boldsymbol{\gamma})$, all orbits tend towards the system bound…

Flux Conjecture. The subgroup $\operatorname{Ham}(M,\omega)$ is $C^1$ close in $\operatorname{Symp}(M,\omega)_0$; equivalently, it is $C^1$-closed there.

Let $(M,\omega)$ be a closed symplectic manifold. For a finite open cover $\mathcal{U}=\{U_i\}$ of $M$, let $e(U_i)$ denote the displacement energy of $U_i$ and set … The Poisson b…

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