Let $(X_i)$ be a sequence of compact metric spaces and $(f_i)$ a sequence of transitive transformations $f_i:X_i \to X_i$ with $0 < h_{top}(f_i) < \infty$.

The sequence of dynamical systems satifies:

- $X_i \subset X_{i+1}$, $h_{top}(f_i) < h_{top}(f_{i+1}) $;
- $X_i$ converges to a compact metric space $X$;
- $f_{i+1}\mid_{X_i} = f_i$ for every $i$;
- Besides, there is a transformation $f:X \to X$ such that f is transitive, $0 < h_{top}(f) < \infty$ and $f\mid_{X_i} = f_i$.
- $h_{top}(f_i)$ converges to $h_{top}(f)$

Assume now that the system $(X_i, f_i)$ is intrinsically ergodic for all $i\ge0$, i.e., it has a unique measure of maximal entropy.

QUESTION. Is $(X,f)$ intrinsically ergodic?

(If it helps, each $(X_i,f_i)$ in my set-up is a transitive subshift of finite type (SFT), but $(X,f)$ is not an SFT.)

If the answer is yes, does there exist a natural way to project the (unique) measure of maximal entropy $\mu$ on $X$ onto $X_i$ so that the projection of $\mu$ is the measure of maximal entropy $\mu_i$ on $X_i$?