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We examine a class of ergodic transformations on a probability measure space 
$(X,{\it\mu})$ and show that they extend to representations of 
${\mathcal{B}}(L^{2}(X,{\it\mu}))$ that are both implemented by a Cuntz family and ergodic. This class contains several known examples, which are unified in our work. During the analysis of the existence and uniqueness of this Cuntz family, we find several results of independent interest. Most notably, we prove a decomposition of 
$X$ for 
$N$-to-one local homeomorphisms that is connected to the orthonormal bases of certain Hilbert modules.
