Abstract
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.
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Notes
Here, \(0_{x}\) is the zero vector in \(T_{x}M\), the tangent space of M at x.
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Muentes Acevedo, J.d.J. On the Continuity of the Topological Entropy of Non-autonomous Dynamical Systems. Bull Braz Math Soc, New Series 49, 89–106 (2018). https://doi.org/10.1007/s00574-017-0049-5
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DOI: https://doi.org/10.1007/s00574-017-0049-5