Uniform tail entropy for real analytic maps

Let $M$ be a compact real analytic manifold of finite dimension. There is a function $a:(0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ with ${\lim }_{t\to 0}a(t)=0$ such that, the tail entropy $h^{\ast }(f,\epsilon )$ of any real analytic map $f$ on $M$ is uniformly bounded above by the scale $a(\epsilon )$.