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Is there an orientation preserving smooth diffeomorphism $f$ on a compact manifold $M$ such that

for every $n\in \mathbb{N}$, there is a smooth diffeomorphism $g:M \to M$, as $n$th root of $f$ (Namely $g^n=f$), but $f$ is not the time-$1$ map of any smooth vector field on $M$?

In the language of group action, an extended version of the question can be restated as follows:

Is there a compact orientable manifold $M$ with an action of $\mathbb{Z}$ by smooth orientation preserving diffeomorphisms on $M$ such that the diffeomorphism $x\to 1.x$ is the time- $1$ map of the flow of no vector field on $M$?

In particular is there a (precise) diffeomorphism $g$ on $2$- torus with $g^2=f$ where $f$ is the linear Anasov diffdeomorphism induced by $\begin{pmatrix}2&1\\1&1 \end{pmatrix}$?

As a related question: Is $SL(2,\mathbb{Z})$ a divisible group?

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    $\begingroup$ Rotation by pi/2 has no square root in SL(2,Z). To see this, notice that if it did have a square root, its eigenvalues would be e^{\pm \pi/4} or e^{\pm 5\pi/4}. But the sums of these potential eigenvalues are not integers. $\endgroup$
    – Anthony Quas
    Commented Feb 21, 2018 at 15:42
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    $\begingroup$ There is a square root of the $\begin{pmatrix}2&1\\1&1 \end{pmatrix}$ map on the torus, namely the $\begin{pmatrix}1&1\\1&0 \end{pmatrix}$ map. It does reverse orientation, and there's no orientation preservering square root, and there are no higher roots, as one can show by an eigenvalue argument. $\endgroup$
    – Lee Mosher
    Commented Feb 22, 2018 at 13:34
  • $\begingroup$ @LeeMosher Thank you and +1 for your interesting comment. But we can not conclude that the Anasov linear flow has no (for example) a 3-th root or more generally a nth root dynamical system. Yes?Is it the time 1 map of a flow? $\endgroup$
    – Ali Taghavi
    Commented Feb 24, 2018 at 12:48
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    $\begingroup$ A diffeomorphism of a smooth manifold $M$ which is the time 1 map of the flow on $M$ generated by a smooth vector field on $M$ is isotopic to the identity on $M$: the flow itself provides the isotopy. But the linear automorphism $g_A$ of the torus defined by a matrix $A \in SL(2,\mathbb{Z})$ is isotopic to the identity if and only if $A$ is the identity matrix, because the matrix $A$ gives the action of $g_A$ on the first homology of the torus. $\endgroup$
    – Lee Mosher
    Commented Feb 24, 2018 at 16:02
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    $\begingroup$ @LeeMosher Thank you.for your interesting comment. so with the same homology argument(with integer coefficients, we conclude that there is no homeomorphism $g$ of torus with $g^3=f$=Anasov diffeomorphism, according to your previous argument in $SL(2, \mathbb{Z}$. $\endgroup$
    – Ali Taghavi
    Commented Feb 24, 2018 at 18:41

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