  For a finite, strongly connected $k$-graph $\Lambda$, an Huef, Laca, Raeburn
and Sims studied the KMS states associated to the preferred dynamics of the
$k$-graph $C^*$-algebra $C^*(\Lambda)$. They found that these KMS states are
determined by the periodicity of $\Lambda$ and a certain Borel probability
measure $M$ on the infinite path space $\Lambda^\infty$ of $\Lambda$. Here we
consider different dynamics on $C^*(\Lambda)$, which arise from a functor $y:
\Lambda \to \mathbb{R}_+$ and were first proposed by McNamara in his thesis. We
show that the KMS states associated to McNamara's dynamics are again
parametrized by the periodicity group of $\Lambda$ and a family of Borel
probability measures on the infinite path space. Indeed, these measures also
arise as Hausdorff measures on $\Lambda^\infty$, and the associated Hausdorff
dimension is intimately linked to the inverse temperatures at which KMS states
exist. Our construction of the metrics underlying the Hausdorff structure uses
the functors $y: \Lambda \to \mathbb{R}_+$; the stationary $k$-Bratteli diagram
associated to $\Lambda$; and the concept of exponentially self-similar weights
on Bratteli diagrams.
