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A remark on inherent differentiablity. (English) Zbl 0689.57021

Summary: Harrison’s analysis of \(C^ r\)-diffeomorphisms which are not conjugate t \(C^ 8\)-diffeomorphisms for \(s>r>0\) is extended to dimension 4. Also topological conjugacy may be generalized to an arbitrary change of differentiable structure. Combining these statements yields: for any smooth manifold of dimension \(\geq 2\) there is a \(C^ r\)-diffeomorphism which is not a \(C^ 8\)-diffeomorphism with respect to any smooth structure.

MSC:

57R50 Differential topological aspects of diffeomorphisms
58C99 Calculus on manifolds; nonlinear operators
57R30 Foliations in differential topology; geometric theory
37-XX Dynamical systems and ergodic theory
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References:

[1] A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl. 11 (1932), 333-375. · JFM 58.1124.04
[2] Jenny Harrison, Unsmoothable diffeomorphisms, Ann. of Math. (2) 102 (1975), no. 1, 85 – 94. · Zbl 0316.57018 · doi:10.2307/1970975
[3] Jenny Harrison, Unsmoothable diffeomorphisms on higher dimensional manifolds, Proc. Amer. Math. Soc. 73 (1979), no. 2, 249 – 255. · Zbl 0405.57019
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