On the Continuity of the Topological Entropy of Non-autonomous Dynamical Systems

Let $M$ be a compact Riemannian manifold. The set $\text{F}^{r}(M)$ consisting of sequences $(f_{i})_{i\in\mathbb{Z}}$ of $C^{r}$-diffeomorphisms on $M$ can be endowed with the compact topology or with the strong topology. A notion of topological entropy is given for these sequences. I will prove this entropy is discontinuous at each sequence if we consider the compact topology on $\text{F}^{r}(M)$. On the other hand, if $r\geq 1$ and we consider the strong topology on $\text{F}^{r}(M)$, this entropy is a continuous map.