$G$ is a function that is periodic in each variable. For star graphs $G$ is defined on ${\mathbb T}^v$ by $$G({{\mathbf x}})=\tan x_1+\cdots+\tan x_v.$$ The equation $$G({{\mathbf x}})=0$$ defines a surface $\Sigma$ embedded in ${\mathbb T}^v$. A flow $\phi^k, k\in{\mathbb R}$ can be defined on ${\mathbb T}^v$ by $$\phi^{k}({{\mathbf x}}_0)={{\mathbf x}}_0+k{{\mathbf L}} \quad\mbox{(mod $\pi$)}.$$

Since $k=0$ is an eigenvalue for star graphs with Neumann boundary conditions considered here, we take ${{\mathbf x}}_0=0$ in this case.

At each value $k=k_n$ we have an intersection of this flow with the surface $\Sigma$. We note that the angle between the normal to the surface $\Sigma$ and the flow $\phi^k$ is given by $$\cos\theta=\frac{|{{\mathbf L}}\cdot\nabla G|}{\|{{\mathbf L}}\|\|\nabla G\|}.$$ For star graphs, $$\nabla G({{\mathbf x}})=(\sec^2 x_1,\ldots,\sec^2 x_v).$$ Hence there exists a constant $c_1>0$ such that $\cos\theta> c_1$. This means that the angle between the flow and the surface $\Sigma$ is uniformly bounded away from 0. We can therefore parameterise $\Sigma$ locally by $v-1$ real variables ${{\boldsymbol \xi}}=(\xi_1,\ldots,\xi_{v-1})$ so that for ${{\mathbf x}}\in\Sigma$, $$x_i=s_i({{\boldsymbol \xi}})$$ and $G(s_1({{\boldsymbol \xi}}),\ldots,s_v({{\boldsymbol \xi}}))=0$.

The central result of Barra and Gaspard is the existence of an invariant measure on the surface $\Sigma$.

\[thm:bar:gas\] Let $f$ be a piecewise continuous function $\Sigma\to{\mathbb 