Let $\mathcal{R}$ be a Markov partition for the cat map. (How) can it be shown that the Lebesgue measure of a rectangle $R_j \in \mathcal{R}$ satisfies $\mu(R_j) = \phi^{-n}$ for some $n$, where $\phi = \frac{1+\sqrt{5}}{2}$?

A "physicist's proof" would be based on the *Ansatz* that $\mathcal{R}$ can be constructed extending local stable and unstable manifolds around the origin *à la* Gallavotti, but it's not clear to me how to build a rigorous argument along these lines.

Any references to work informing an answer would be particularly appreciated. Best of all would be a pointer accounting for the relative measures and multiplicities of *all* rectangles.