Let $G$ be a countable amenable group and let $X,Y$ be subshifts with finite alphabet over $G$. Suppose that $h(X) = h(Y)$ (equal topological entropy). I am interested in continuous factor maps $\pi: X \to Y$ that preserve topological pressure under pullback, in the sense that for a reasonably regular continuous potential $f: Y \to \mathbb{R}$, $P_X(f \circ \pi) = P_Y(f)$. Are there sufficient conditions known on $X,Y, \pi$ that imply this?
One sufficient condition would be for the pushforward by $\pi$ to preserve the entropy of every invariant measure on $X$. This would happen if $\pi$ were finite-to-one, or if $Y$ were uniquely ergodic, but less restrictive hypotheses would be nice.
