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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.

In this paper we extend the definitions of mean dimension and metric mean di-mension for non-autonomous dynamical systems. We show some properties of this extension and furthermore some applications to the mean dimension and metric mean dimension of single continuous maps.

Anosov families were introduced by A. Fisher and P. Arnoux motivated by generalizing the notion of Anosov diffeomorphism defined on a compact Riemannian manifold. They are time-dependent dynamical systems with hyperbolic behavior. In addition to presenting several properties and examples of Anosov families, in this paper we build local stable and local manifolds for such families. © Hokkaido University.