181 problems

An integrable Hamiltonian system is a Hamiltonian system with more than two degrees of freedom whose dynamics is integrable. Consider such a system subjected to a small Hamiltonian…

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…

Arnol'd's conjecture. Every Hamiltonian diffeomorphism $\varphi$ possesses at least as many fixed points as a smooth function $f:M\to\mathbb{R}$ possesses critical points.

Let $\mathcal{T}_0$ be an analytic Lagrangian Diophantine quasi-periodic torus invariant under a real-analytic Hamiltonian system. A set of tori is accumulated by $\mathcal{T}_0$ i…

Rabinowitz's minimal periodic solution conjecture. Under these hypotheses, for every prescribed positive period, the system possesses a non-constant solution having that period as…

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.

Nekhoroshev stability conjecture. Nekhoroshev's theorem actually holds on the set $\mathcal{P}^{\bullet}$.

In a real-analytic, nearly integrable Hamiltonian system with three or more degrees of freedom, consider the relative measure of the phase space that is free of invariant tori. Arn…

Herman's conjecture. The set of non-wandering points for the flow of the $N$-body problem is nowhere dense on every energy level for $N\geq 3$.

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 semipositive symplectic manifold, let $\phi:M\to M$ be a Hamiltonian diffeomorphism with finitely many periodic points, and let $\Lambda_{\mathbb{Q}}$ be the…

Let an elliptic equilibrium point be an equilibrium of a Hamiltonian system, with quadratic part of the Hamiltonian function at the equilibrium point. The quadratic part is sign-de…

Hofer's conjecture. If the path generated by $H$ is not length minimizing in its homotopy class, then the Hamiltonian flow has a nonconstant contractible periodic orbit with period…

A symplectic manifold is a manifold $(M,\omega)$ equipped with a symplectic form, and a Hamiltonian diffeomorphism is the time-one map of a Hamiltonian isotopy. A Hamiltonian diffe…

Let $(M,\omega)$ be a closed symplectic manifold, let $L$ be a closed Lagrangian submanifold of $M$, and let $\varphi\in \operatorname{Ham}(M,\omega)$ be a Hamiltonian diffeomorphi…

Arnol'd's conjecture. A general Hamiltonian should have a dense orbit on a general energy surface.

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…

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…

Polterovich–Shelukhin conjecture. For every closed symplectic manifold $(M,\omega)$,

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…

Arnold diffusion concerns global instability in Hamiltonian systems under arbitrarily small perturbations, with orbits undergoing large effects over time. In his example, Arnold ex…

Let $(M,\omega)$ be a closed symplectic manifold, let $k\geq 2$ be an integer, and equip the group of Hamiltonian diffeomorphisms with Hofer's metric. Hofer-distance conjecture. Th…

Positive measure conjecture. There exist constants $c,C>0$ independent of $\varepsilon$ such that

A selector is a minimax-type invariant assigned to a Hamiltonian, and a symplectic manifold is said here to admit a nice Floer homology when its Floer homology has the properties r…

Let $M$ be a manifold of dimension $n$, let $\mathcal{Z}$ be the perturbation space considered in the surrounding argument, and let $\pi$ denote its projection onto $sp(n-1)$. Surj…