# Problem 63

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$C^r$ diffeos that are not $C^{r+1}$ qualitatively. Find a $C^1$ diffeo $f : V \to V$ and a $C^1$ embedding $g:V \to M$ and $\tilde f$ extending $f$ to $M$ such that $\tilde f$ is $C^2$ on $M$, but $\tilde f |V$ is qualitatively not $C^2$ (i.e. the qualitative behavior of $\tilde f|V$ is due to the irregularity of $V$ but not of $\tilde f$)

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### FL

A bit of interpretation here, I am not sure I understand well the question.

### There are Cr diffeomorphisms

There are Cr diffeomorphisms that are not Cr conjugated to a Cr+1 diffeomorphism

J. Harrison, Unsmoothable diffeomorphisms. Ann. of Math. 102 (1975), 85–94.

There exists a Cdiffeomorphism of the annulus which preserves a Lipschitz circle supporting a Denjoy counterexample.

G. Hall, Bifurcation of an attracting invariant circle: a Denjoy attractor. Ergodic Theory Dynam. Systems 3 (1983), 87–118.