Let $F$ be a smooth manifold and let $f:F\to F$ be a diffeomorphism. The map
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\begin{align*}
\pi:T_f&\to S^1, & \pi([y,t])&=e^{2\pi i t}
\end{align*}
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is a smooth fibre bundle with fibre $F$. Its transition functions may be chosen to take values in the cyclic subgroup of $\operatorname{Diff}(F)$ generated by $f$, or in any Lie subgroup of $\operatorname{Diff}(F)$ containing $f$ when such a subgroup has been specified.