Ergodic extensions and Hilbert modules associated to endomorphisms of MASAS

We show that a class of ergodic transformations on a probability measure space $(X,\mu )$ extends to a representation of $\mathcal{B}(L^2(X,\mu ))$ that is both implemented by a Cuntz family and ergodic. This class contains several known examples, which are unified in this work. During the analysis of the existence and uniqueness of such a Cuntz family we give several results of individual interest. Most notably we prove a decomposition of $X$ for $N$-to-one local homeomorphisms that is connected to the orthonormal basis of Hilbert modules. We remark that the trivial Hilbert module of the Cuntz algebra ${\mathcal{O}}_N$ does not have a well-defined Hilbert module basis (moreover that it is unitarily equivalent to the module sum ${\sum }_{i=1}^n{\mathcal{O}}_N$ for infinitely many $n\in \mathbb{N}$).