R}$. Then $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(k_n{{\mathbf L}})=\int_{\Sigma} 
f({{\boldsymbol \xi}}){{\mathrm d}}\nu({{\boldsymbol \xi}})
\label{eq:bar:gas}$$ where the measure $\nu$ is given by $${{\mathrm d}}\nu({{\boldsymbol \xi}})=\frac{J({{\boldsymbol \xi}}){{\mathrm d}}{{\boldsymbol \xi}}}
{\int_{\Sigma}J({{\boldsymbol \xi}}){{\mathrm d}}{{\boldsymbol \xi}}}$$ and ${{\mathrm d}}{{\boldsymbol \xi}}={{\mathrm d}}\xi_1\cdots{{\mathrm d}}\xi_{v-1}$ is Lebesgue measure. $J$ is the Jacobian determinant $$J({{\boldsymbol \xi}})=\left|
  \begin{array}{ccc}
L_1 & \cdots & L_v \\
\frac{\partial s_1}{\partial \xi_1} & \cdots &
\frac{\partial s_v}{\partial \xi_1}\\
\vdots & \ddots & \vdots  \\
\frac{\partial s_1}{\partial \xi_{v-1}} & \cdots &
\frac{\partial s_v}{\partial \xi_{v-1}}
\end{array} \right|$$

For completeness, we sketch a proof of theorem \[thm:bar:gas\] for star graphs with $v$ bonds.

[*Proof.*]{}

Let $\tilde{f}:{\mathbb T}^v\to{\mathbb R}$ be an extension of $f$ to ${\mathbb T}^v$, so that $\tilde{f}\big|_{\Sigma}=f$, i.e.$$\tilde{f}({{\mathbf x}})=f({{\mathbf x}})\qquad\mbox{for all ${{\mathbf x}}\in\Sigma$.}$$ We let $\tilde{f}$ be constructed in such a way that for all ${{\boldsymbol \xi}}\in\Sigma$, $\tilde{f}(\phi^{k}({{\boldsymbol \xi}}))$ is a differentiable function of $k$ with compact support in some neighbourhood of $k=0$.

Let $\epsilon>0$. We construct the set $\Sigma_{\epsilon,{{\mathbf L}}}$ which is a thickening of $\Sigma$ in the direction of the flow $\phi^k$, $$\Sigma_{\epsilon,{{\mathbf L}}}:=\{{{\mathbf x}}\in{\mathbb T}^v :\exists{{\boldsymbol \xi}}\in
\Sigma, k\in[-\epsilon,\epsilon]:{{\mathbf x}}=\phi^k({{\boldsymbol \xi}})\}\subseteq
{\mathbb T}^v.$$ We define ${{1\!\!1}}_A$, the indicator function of a set $A$, by $${{1\!\!1}}_A(x):=\left\{
  \begin{array}{lr}
    1, & x\in A\\
    0, & x\not\in A
  \end{array}\right.$$ The indicator function 