Point basins, equilibrium-set basins and aggregate volume

Assume countable choice. Let $V$ be a nonempty finite set, $T_V={\mathbb{R}}^V\mathrm{/}(2\pi \mathbb{Z}{)}^V$ the phase torus with its flat distance $d$, and $\mu$ its normalized phase volume. Let $\mathrm{\Phi }:\mathbb{R}\times T_V\to T_V$ be a complete flow: it is continuous, ${\mathrm{\Phi }}_0$ is the identity, and ${\mathrm{\Phi }}_{t+s}={\mathrm{\Phi }}_t\circ {\mathrm{\Phi }}_s$. An equilibrium $q$ satisfies ${\mathrm{\Phi }}_t(q)=q$ for every real $t$. Its point basin is $B(q)=\{x\in T_V:d({\mathrm{\Phi }}_t(x),q)\to 0\text{ as }t\to \mathrm{\infty }\}$.

For a nonempty invariant set $C\subseteq T_V$, meaning ${\mathrm{\Phi }}_t(C)=C$ for every real $t$, put $d(x,C)={\inf }_{c\in C}d(x,c)$. Its attraction set is $B(C)=\{x\in T_V:d({\mathrm{\Phi }}_t(x),C)\to 0\text{ as }t\to \mathrm{\infty }\}$. Both $B(q)$ and $B(C)$ are phase-measurable. In particular, $\mu (B(q))$ and $\mu (B(C))$ are well-defined probabilities under the uniform phase law.

For a specified family of equilibria $Q$, its aggregate point-basin volume means $\mu ({\bigcup }_{q\in Q}B(q))$ when that union is phase-measurable. This is a measure of the union; an uncountable sum of point-basin volumes is not implied. If $Q=C$ consists of equilibria, the union is contained in $B(C)$, but equality requires an additional point-convergence assertion. Stability of $C$, including a rotation orbit of equilibria, alone does not supply point attraction. For a family of graphs, phase volume is normalized separately on the phase torus associated with each graph's vertex set. An almost-everywhere or probability-one attraction conclusion requires an argument establishing the relevant attraction and exceptional-set claim.

Point stability, attraction and stability of a set

Finite phase tori, local lifts and the flat metric

Normalized phase volume and almost-everywhere assertions

Probability measures and probability spaces

The Axiom of Countable Choice (AC_ω)

Given: The phase space, measure, complete flow, equilibrium $q$, and nonempty invariant set $C$ in the definition.

Goal: Verify the measurability and volume conventions and the stated distinction between set and point attraction.

1.1 For any nonempty $C$, the triangle inequality gives $\mathrm{\mid }d(u,C)-d(v,C)\mathrm{\mid }\le d(u,v)$: apply $d(u,c)\le d(u,v)+d(v,c)$ for every $c\in C$, take the infimum, and exchange $u,v$. Thus $x\mapsto d({\mathrm{\Phi }}_t(x),C)$ is continuous for fixed $t$, and $t\mapsto d({\mathrm{\Phi }}_t(x),C)$ is continuous for fixed $x$. The same holds for $d({\mathrm{\Phi }}_t(x),q)$ by taking $C=\{q\}$. Hence their sublevel sets with a weak inequality are closed. [F1, algebra]

1.2 Let $D=\{k{2}^{-j}:k,j\in {\mathbb{N}}_0\}$. By [F5], ${\mathbb{N}}_0\times {\mathbb{N}}_0$ is at most countable. Listing its pairs and applying $(k,j)\mapsto k{2}^{-j}$ lists all of $D$, with repetitions allowed, so $D$ is countable. Also $2^j\ge j+1$ for every $j\ge 0$: it holds at $j=0$, and the induction step is $2^{j+1}\ge 2(j+1)\ge j+2$. Given $\eta >0$, choose $m\ge 1$ with $1\mathrm{/}m<\eta$ by [F6]; for $j\ge m$ we have $2^{-j}\le 1\mathrm{/}(j+1)\le 1\mathrm{/}m<\eta$. Thus $2^{-j}\to 0$. [F5, F6, F7, algebra]

2.1 For $t\ge 0$, let $k_j$ be the integer part of $2^{j}t$. Then $k_j\ge 0$ and $r_j=k_j{2}^{-j}\in D$ satisfies $0\le t-r_j<2^{-j}$, so $r_j\to t$ by step 1.2. This proves density in $[0,\mathrm{\infty })$. Moreover, if $t\ge N$ for an integer $N\ge 0$, the integer $2^{j}N$ is at most $2^{j}t$, so $k_j\ge 2^{j}N$ and $r_j\ge N$. Hence the approximating dyadics can always be taken in the same tail $[N,\mathrm{\infty })$. [F4, step 1.2, algebra]

3.1 For $m\ge 1$, $N\in {\mathbb{N}}_0$ and $r\in D$, write $E_{m,r}=\{x:d({\mathrm{\Phi }}_r(x),C)\le 1\mathrm{/}m\}$. Then $B(C)={\bigcap }_{m\ge 1}{\bigcup }_{N\ge 0}{\bigcap }_{r\in D,r\ge N}{E}_{m,r}$. Indeed, convergence implies the right side: for each $m$, choose a real threshold $T\ge 0$ after which the distance is smaller than $1\mathrm{/}m$, and take $N=\lfloor T\rfloor +1$ using [F4]. Conversely, if $x$ is in the right side, fix $m$ and a corresponding $N$. For every real $t\ge N$, step 2.1 supplies dyadics $r_j\ge N$ tending to $t$. The weak bounds $d({\mathrm{\Phi }}_{r_j}(x),C)\le 1\mathrm{/}m$ pass to $t$ by the time continuity in step 1.1. Given $\epsilon >0$, choose $m$ with $1\mathrm{/}m<\epsilon$ by [F6]; the resulting bound for all $t\ge N$ proves convergence. [F4, F6, step 1.1, step 2.1, algebra]

4.1 Each $E_{m,r}$ is closed, hence Borel. The displayed expression is Borel by countable sigma-algebra closure and therefore phase-measurable, since the phase-measurable sigma-algebra contains all open sets. Repeating the argument with $C=\{q\}$ proves measurability of $B(q)$. Their volumes are consequently defined by the probability measure $\mu$. [F2, F3, step 1.1, step 1.2, step 3.1]

5.1 If $q\in C$, then $d(y,C)\le d(y,q)$ for every $y$, so $B(q)\subseteq B(C)$ and the same inclusion holds for the union over $q\in C$. Inclusion alone does not establish equality or measurability of an arbitrary uncountable union; the aggregate convention explicitly requires the latter. For the identity flow on the phase torus, every point is a stable equilibrium and the entire torus is a stable invariant set, while $B(q)=\{q\}$: set stability therefore does not imply attraction to any prescribed point. The separate normalization and almost-everywhere conventions are those of phase volume and probability, and no convergence conclusion follows from their definitions alone. [F1, F2, step 4.1, algebra] ∎