Gradient trajectories and complete compact phase dynamics

Assume countable choice. Let $V$ be a nonempty finite set, let $T_V={\mathbb{R}}^V\mathrm{/}(2\pi \mathbb{Z}{)}^V$ have its flat metric, and let $U:T_V\to \mathbb{R}$ be smooth, meaning that its expressions in angular charts have derivatives of every order. Its gradient $\mathrm{grad}U$ is the vector field satisfying $⟨\mathrm{grad}U(q),v⟩=d{U}_q(v)$ for every tangent vector $v$ at $q$.

Equip $\mathbb{R}\times T_V$ with the product smooth structure: for an angular chart $\kappa :O\to B$ and an open interval $J\subseteq \mathbb{R}$, its product chart on $J\times O$ is $(t,p)\mapsto (t,\kappa (p))\in J\times B$. A map between smooth manifolds is $C^2$ when it is continuous and its coordinate representatives in source and target charts have continuous partial derivatives of all orders at most two. Joint $C^2$ regularity of a map on $\mathbb{R}\times T_V$ uses these product charts. A $C^2$ diffeomorphism is a bijection whose map and inverse are both $C^2$.

A downward gradient trajectory through $q_0\in T_V$ is the maximal integral curve $q:I\to T_V$ satisfying $q(0)=q_0$ and $\dot{q}(t)=-\mathrm{grad}U(q(t))$. Here an integral curve is a differentiable curve whose velocity equals the indicated vector field, and maximal means that it has no extension as such a curve to a strictly larger interval. In this setting $I=\mathbb{R}$. Define $\mathrm{\Phi }(t,q_0)={\mathrm{\Phi }}_t(q_0)=q(t)$. The map $\mathrm{\Phi }:\mathbb{R}\times T_V\to T_V$ is jointly $C^2$, ${\mathrm{\Phi }}_0$ is the identity, and ${\mathrm{\Phi }}_{t+s}={\mathrm{\Phi }}_t\circ {\mathrm{\Phi }}_s$; consequently each ${\mathrm{\Phi }}_t$ is a $C^2$ diffeomorphism with inverse ${\mathrm{\Phi }}_{-t}$. These are properties of the mathematical ODE.

Given: The data in Facts & Assumptions.

Goal: Establish that the trajectory and flow in the definition are well defined and have the stated regularity and group law.

1.1 By [F1] and [F2], $X=-\mathrm{grad}U$ is a smooth vector field on the compact smooth manifold $T_V$. In an angular chart its coordinate field is $C^2$. By [F4], local solutions exist uniquely and depend jointly $C^2$ on time and initial coordinates. On chart overlaps the coordinate descriptions represent the same vector field, so uniqueness makes these local solutions agree wherever their domains overlap. [F1, F2, F4, given]

2.1 By [F3], each maximal integral curve of $X$ has domain $\mathbb{R}$; uniqueness in step 1.1 makes the curve through each $q_0$ unique. Thus $\mathrm{\Phi }$ is well defined. To see joint $C^2$ regularity near any $(t_0,q_0)$, cover the compact trajectory segment between times $0$ and $t_0$ by finitely many chart neighborhoods and subdivide the segment into finitely many short intervals on which [F4] applies. For initial points near $q_0$ and times near $t_0$, the resulting local flow maps compose to give $\mathrm{\Phi }$; their chart transitions and compositions are $C^2$. The product-chart transitions are locally $(t,u)\mapsto (t,u+\lambda )$ for a fixed $\lambda \in (2\pi \mathbb{Z}{)}^V$, hence are smooth. Thus the local representatives just constructed give continuity and joint $C^2$ regularity on the product manifold in the sense of [F5]. The same argument applies for negative $t_0$. [F1, F3, F4, F5, step 1.1]

3.1 The curve $t\mapsto {\mathrm{\Phi }}_t({\mathrm{\Phi }}_s(q))$ and the curve $t\mapsto {\mathrm{\Phi }}_{t+s}(q)$ both solve the autonomous equation and agree at $t=0$. Local uniqueness, applied along their common real time interval, makes them equal. Hence ${\mathrm{\Phi }}_{t+s}={\mathrm{\Phi }}_t\circ {\mathrm{\Phi }}_s$ and ${\mathrm{\Phi }}_0$ is the identity. Taking $s=-t$ in both orders gives inverse maps ${\mathrm{\Phi }}_t$ and ${\mathrm{\Phi }}_{-t}$; their $C^2$ regularity follows from step 2.1. Hence they are $C^2$ diffeomorphisms in the sense of [F5]. [F4, F5, step 1.1, step 2.1] ∎