Let $X$ be a compact metric space and $f:X \to X$ a continuous map. We say that $(X,f)$ is approximated from below by a sequence of compact metric spaces $(X_i)_{i \geq 1}$ and a sequence of continuous transformations $(f_i)_{i \geq 1}$ on $X_i$ if we have:
- $X_i \subset X_{i+1}$ for every $i \geq 1$;
- $X = \overline{\mathop{\bigcup}\limits_{i=1}^{\infty}X_i}$;
- $f_{i+1}|_{X_i} = f_i$, $f|_{X_i} = f_i$ for every $i \geq 1$.
And we say that $(X,f)$ is approximated from above by a sequence of compact metric spaces $(Y_i)_{i \geq 1}$ and a sequence of continuous transformations $(g_i)_{i \geq 1}$ on $Y_i$ if we have:
- $Y_{i+1} \subset Y_i$ for every $i \geq 1$;
- $X = \mathop{\bigcap}\limits_{i=1}^{\infty}Y_i$;
- $g_{i}\mid_{Y_{i+1}} = g_{i+1}$, $f\mid_{Y_i} = g_i$ for every $i \geq 1$.
Assume now that $X$ is approximated from above by a sequence $(X_i,f_i)_{i \geq 1}$ and from below by $(Y_i, g_i)_{i \geq 1}$ and $(X_i,f_i)$ and $(Y_i, g_i)$ are intrinsically ergodic systems for all $i \geq 1$, i.e., each of them has a unique measure of maximal entropy.
QUESTIONS:
- Is $(X,f)$ intrinsically ergodic?
- Is there any example of a dynamical system approximated from above by intrinsically ergodic systems that is not intrinsically ergodic? In February 2012 I asked a question about limits on intrinsically ergodic systems (see link attached) Limits of intrinsically ergodic systemsLimits of intrinsically ergodic systems, the answer gave me a counterexample for spaces approximated from below.
(If it helps, each $(X_i,f_i)$ and each $(Y_i, g_i)$ in my set-up is an intrinsically ergodic subshift of finite type for every $i$).
