A null basin for all strict saddles of a smooth compact gradient flow

On an overlap of angular charts the transition is locally $y=x+a$ for a constant lattice vector $a$. If $u_x$ and $u_y$ are the expressions of $U$, then $u_y(y)=u_x(y-a)$ and $\mathrm{\partial }{u}_y\mathrm{/}\mathrm{\partial }{y}_v=\mathrm{\partial }{u}_x\mathrm{/}\mathrm{\partial }{x}_v$ at corresponding points. The transition derivative is the identity, so the gradient components in the Statement agree as tangent vectors and define a global $C^1$ field. Their Euclidean inner product with a vector $w$ is ${\sum }_{v\in V}(\mathrm{\partial }U\mathrm{/}\mathrm{\partial }{x}_v)w_v=dU(w)$, so this is the flat-metric gradient. Its downward field is $-\mathrm{\nabla }U$ in each angular chart. By [F2], uniqueness patches the local flows along each trajectory, and a finite subdivision of any compact time interval gives $C^1$ dependence on the initial point. Completeness and uniqueness give ${\mathrm{\Phi }}_{t+s}={\mathrm{\Phi }}_t\circ {\mathrm{\Phi }}_s$ and ${\mathrm{\Phi }}_t^{-1}={\mathrm{\Phi }}_{-t}$, so each time map is a $C^1$ diffeomorphism. At a critical point $p$, the constant trajectory is the unique trajectory through $p$. In an angular chart the variational equation there is $Y^{\mathrm{\prime }}=-H_{p}Y$, $Y(0)=I$; consequently $D{\mathrm{\Phi }}_1(p)=e^{-H_p}$ by [F2].

Fix a strict saddle $p$. If $H_{p}v=\lambda v$ with $\lambda <0$ and $v\mathrm{\ne }0$, a real nonzero eigenvector may be chosen because $H_p$ and $\lambda$ are real. The exponential series gives $D{\mathrm{\Phi }}_1(p)v=e^{-\lambda }v$, with $e^{-\lambda }>1$. Apply [F3] to the coordinate representation of ${\mathrm{\Phi }}_1$ at $p$, shrinking its coordinate domain as necessary. Its trapping neighbourhood $B_p$ is open in the torus, and its embedded disc $W_p$ has dimension strictly less than $\mathrm{\mid }V\mathrm{\mid }$, since the unstable generalized eigenspace contains $v$. Thus every point whose nonnegative integer iterates stay in $B_p$ belongs to $W_p$.

Let $S$ be the set of strict saddles. The angular boxes with rational endpoints form a countable base for $T_V$, so their intersections with $S$ form a countable base for $S$. The sets $B_p\cap S$, $p\in S$, cover $S$. By [F4], choose a countable subcover $B_{p_j}\cap S$, with its associated discs $W_{p_j}$. If $S$ is empty, the desired basin is empty. Otherwise, for any initial point $x$ converging to $p\in S$, choose $j$ with $p\in B_{p_j}$. Since $B_{p_j}$ is open, there is an integer $N\ge 0$ such that ${\mathrm{\Phi }}_{N+k}(x)\in B_{p_j}$ for every integer $k\ge 0$. By step 2.1, ${\mathrm{\Phi }}_N(x)\in W_{p_j}$. Therefore the basin is contained in ${\bigcup }_{j,N\ge 0}{\mathrm{\Phi }}_{-N}(W_{p_j})$.

Each $W_{p_j}$ is the $C^1$ image of an open subset of Euclidean space of dimension less than $\mathrm{\mid }V\mathrm{\mid }$, so [F4] makes it manifold-null. By step 1.1 and [F4], every ${\mathrm{\Phi }}_{-N}(W_{p_j})$ is manifold-null; countable choice makes their countable union, and every subset of it, manifold-null. Cover the torus by countably many angular charts. Since $\mathrm{\mid }V\mathrm{\mid }\ge 1$, the definition in [F4] makes the coordinate image of the basin intersected with each chart domain elementary Euclidean-null. In each chart, [F5] converts this coordinate nullity to a phase-null subset. Their countable union is phase-null by completeness and countable additivity of phase volume. This proves the claim.