9 problems

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…

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…

Let $h$ be an integrable Hamiltonian on the action space, let $P$ be a generic perturbation, and consider the nearly integrable Hamiltonian … For any two points $y'$ and $y”$ on th…

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 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 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…