In this paper we study the preservation of various notions of expansivity in discrete dynamical systems and the induced map for $n$-fold symmetric products and hyperspaces. Then we give a characterization of a compact metric space admitting hyper $N$-expansive homeomorphisms via the topological dimension. More precisely, we show that $C^0$-generically, any homeomorphism on a compact manifold is not hyper $N$-expansive for any $N\in \mathbb{N}$. Also we give some examples to illustrate our results.
In this paper we study the preservation of various notions of expansivity in discrete dynamical systems and the induced map for $n$-fold symmetric products and hyperspaces. Then we give a characterization of a compact metric space admitting hyper $N$-expansive homeomorphisms via the topological dimension. More precisely, we show that $C^0$-generically, any homeomorphism on a compact manifold is not hyper $N$-expansive for any $N\in \mathbb{N}$. Also we give some examples to illustrate our results.
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