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There is a considerable amount of research in dynamics which aims at the understanding of the generic behaviour of the generic diffeomorphism. I do believe that people care about any class of differentiability, but most of the techniques developed so far do not reach much beyond the $C^1$-topology. Tipically one has to deform a given diffeo to arrive at generic conditions and it is much much harder to make small perturbations in the $C^{\infty}$-topology than in the $C^1$-topology. An arquetypical archetypical example is the $C^1$-closing lemma which is not known to hold in finer topologies.

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There is a considerable amount of research in dynamics which aims at the understanding of the generic behaviour of the generic diffeomorphism. I do believe that people care about any class of differentiability, but most of the techniques developed so far do not reach much beyond the $C^1$-topology. Tipically one has to deform a given diffeo to arrive at generic conditions and it is much much harder to make small perturbations in the $C^{\infty}$-topology than in the $C^1$-topology. An arquetypical example is the $C^1$-closing lemma which is not known to hold in finer topologies.