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 $r\ge 1$ and we consider the strong topology on ${\text{F}}^r(M)$, this entropy is a continuous map.