Are analytic maps $h$-expansive?
See Buzzi, Jérôme. Intrinsic ergodicity of smooth interval maps. Israel J. Math. 100 (1997), 125--161. And also Yomdin, Y. Volume growth and entropy. Israel J. Math. 57 (1987), no. 3, 285--300.
M. Misiurewicz  built $C^r$ diffeomorphisms without measures maximizing the entropy. These were the first known examples of non entropy-expansive maps. Nowadays, it is easy to construct a real-analytic example, e.g., the map $f:[-1,1]\to[-1,1]$, $x\mapsto 1-2x^2$. Indeed, for any $\epsilon>0$, the "entropy below scale $\epsilon$" is positive. This can be shown by considering an orbit visiting $(-\epsilon/2,-\epsilon/4)\cup(\epsilon/4,\epsilon/2)$ with positive frequency.
On the other hand, the weaker property of asymptotic entropy-expansiveness, i.e., that the "entropy below scale $\epsilon$" converges to zero as $\epsilon$ goes to $0$, holds for every $C^\infty$-map of a compact manifold and in particular for any real-analytic map . For more precise results, see  and .