Let $(X, \mathcal{B}, \mu, S)$ and $(Y, \mathcal{C}, \nu, T)$ be invertible probability-measure-preserving systems, with a measurable factor map $\pi: X \to Y$, i.e. $\pi \circ S = T \circ \pi$. Suppose that there is some $N \in \mathbb{N}$ such that $\nu$-almost every $y \in Y$ has at most $N$ preimages under $\pi$, i.e. $\# \pi^{-1}(\{y\}) \leq N$. If necessary, I'm happy to assume that $X$ and $Y$ are compact metric spaces, $\mu$ and $\nu$ are Borel, and $\pi, S, T$ are continuous.
I'm trying to show that $\pi$ preserves measure-theoretic entropy, i.e. $h(\nu) = h(\mu)$. I'd like to apply the Abramov-Rokhlin entropy formula, which expresses the entropy of a skew product transformation as the sum of the entropy of the base and the entropy of the fibre. To do this, I'd like to realize $X$ as the direct product $Y \times S$ (or a subset of this), with $F = \{ 1, \dots, N \}$, together some probability vector $\mathbf{p} \in \mathbb{R}^N$, such that $\mu = \nu \times \mathbf{p}$. I'd then try to realize $S$ as $S(y,k) = (T(y), \alpha(y)(k))$ where for each $y$, $\alpha(y)$ is an invertible measure-preserving transformation of $(F, \mathbf{p})$.
The problem is that if the $\alpha$'s give a transitive action of $S_N$, this would require that $\mathbf{p}$ be uniform. In particular, it seems that the fiber cardinality would have to be constant, not just bounded. Even forgetting about the measure, this is an issue, because the $\alpha$'s would have to be bijections.
Two ideas I've played with:
The number of preimages $\# \pi^{-1}(\{y\})$ is constant along the orbit of $y$, so I could try partitioning $X$ and $Y$ into at most $N$ disjoint subsystems on which $\pi$ is constant-to-one, then applying the Abramov-Rokhlin formula separately on each one, but I don't know how to show that this decomposition would be measurable.
Bögenschutz and Crauel (1990) have a generalization of the Abramov-Rokhlin formula for a skew product transformation (with the notation above) in which we no longer require a fixed measure on the fiber, but only that $\mu$ be invariant with fixed $Y$-marginal $\nu$. This might solve the problem by letting me add a null set of fixed points to $X$, for instance, to make $\pi$ constant-to-one.
I'm also hoping to apply this in the case where $S$ and $T$ are really actions of a countable amenable group $G$, using Ward and Zhang's generalization of the Abramov-Rokhlin formula. I haven't worked through the proofs to see if the Bögenschutz-Crauel and Ward-Zhang generalizations are compatible with each other.
