22 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…

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…

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

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

Consider a perturbed integrable Hamiltonian with action variables $I$ and perturbation $H_1(I,\ph,\eps)$, where $\ph$ denotes the angle variables and $\eps$ is sufficiently small.…

Arnold's conjecture. The mechanism of instability based on the existence of transition chains is applicable to the general case, for example to the problem of three bodies.

Let $n\geq 3$ and let ${{\bf m}{\alpha}}_1,\ldots,{{\bf m}{\alpha}}_{n-1}$ denote the steepness indices of the Hamiltonian. Let $a$ be the exponent governing the long-time stabilit…

A mechanical system consists of a rotator and a pendulum coupled by a small, time-periodic Hamiltonian perturbation; Arnold diffusion refers to diffusing orbits along which the rot…

Define the bounded-motion region … where $R$ is the Kerr radial potential, $r^*$ is the local maximum of $-R$, and $r_*$ is the local minimum of $-R$ to its right. Generic Kerr bou…

Let $N_\varepsilon$ be the perturbed normally hyperbolic invariant manifold associated with the Kerr photon shell, and let $(a,M,E,Q,L_z)$ satisfy the conditions in assumption (2)…

Define the bounded-motion region … Here $E$ is the particle energy, $L$ its angular momentum, $V$ the effective potential, and $H$ the Hamiltonian. Generic Schwarzschild bounded-ph…

Let $N_\varepsilon$ denote the perturbed normally hyperbolic invariant manifold associated with the photon sphere, and let $L$ and $L_z$ be the angular-momentum quantities used in…

Consider a Hamiltonian system subject to a small time-periodic perturbation, and let the energy of an orbit be the quantity whose evolution is observed. Quantitative properties of…

Consider the near-integrable, time-periodic a priori unstable Hamiltonian … where $y\in\overline{\mathcal D}\subset\mathbb{R}^n$, $x\in\mathbb{T}^n$,…

Consider Arnold's a priori unstable Hamiltonian … where $q,\varphi,t\in\mathbb{T}$ are angles, $p,I\in\mathbb{R}$ are actions, $H_0(p,q)=\frac{p^2}{2}+\cos q-1$, and…

Arnold diffusion conjecture. For sufficiently large $r\in{\mathbb N}\cup\\{\infty,\omega\\}$ and an open dense set of $C^r$ perturbations: (A) diffusion exists; (B) the projection…

Finite-regularity Arnold diffusion conjecture. There exists $r_0>0$ such that, for every $n\geq 2$, every $\gamma>0$, and every $r$ with $r_0\leq r<\infty$, a cusp residual set of…

Fermi's conjecture. Quasi-ergodicity is a generic property of Hamiltonian systems.

Let $D_{E_Z}$ and $D_{J_F}$ be the diffusion coefficients measured in the transformed canonical variables, and let $\lVert R_{opt}\rVert$ denote the optimal normal-form remainder.…

Diffusion-time conjecture. For the orbits constructed in this paper, the diffusion time $T$ can be chosen to satisfy

Let a Hamiltonian system be integrable when it admits the integrable structure under consideration, and consider perturbations in the complement of the integrable systems. Instabil…

Arnold diffusion conjecture. Arnold diffusion due to the Arnold mechanism is present quite generically, for instance in the three-body problem.