On $\mathbb{T}^n$ with a Riemannian metric, the stable norm is defined as 
$$\Vert h\Vert=\inf \sum |r\_i| \cdot \mathrm{length}(\sigma\_i),$$


where $h\in H\_1(\mathbb{T}^n,\mathbb R)$ and $\sum\_i r\_i\sigma\_i$ is a Lipschitz cycle representing $h$. The $\inf$ is taken over all such cycles. 


The question is, if we consider 


(1)$h$ is completely rational, i.e. there are n-1 linearly independent integer vectors perpendicular to $h$, 


(2) The $\inf$ is achieved on a set supporting a measure that is uniquely ergodic. (This excludes Hedlund's counterexample)


then is it correct that the $\inf$ is always achieved by a cycle, i.e. the measure is supported on a closed orbit? 


This is true in 2-dim case, but not sure about the high dim case. 



rational rotation vector and closed curves
Let $\mathbb T^n=\mathbb R^n\pmod {2{\cdot}\pi}$. 
Consider the embedding $f\colon\mathbb T^2\hookrightarrow \mathbb T^3$
 defined as
 $$f\colon(u,v)\mapsto (\sin u,\cos u, v).$$


It is easy to find a metric on $\mathbb T^3$ so that for $h=(0,0,1)$,
 the infimum is achieved on a geodesic which runs in the embedded $\mathbb T^2$ in an irrational direction.



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> Is that what you wanted? 
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