The suspension of a circle diffeomorphism: leaves and return germs

Leaf type. The period set $K_y=\{k\in \mathbb{Z}:f^k(y)=y\}$ is a subgroup of $\mathbb{Z}$. If the orbit of $y$ is periodic of minimal period $k$, then $K_y=k\mathbb{Z}$ with $k\ge 1$ minimal, and $L_y\cong \mathbb{R}\mathrm{/}k\mathbb{Z}\cong S^1$ by [F1] and [F3]; if the orbit is not periodic, $K_y=\{0\}$ and $L_y\cong \mathbb{R}$. This proves claim 1.

The generating loop and its germ. If $y$ has minimal period $k$, the path $a(t)=[(kt,y)]$ is closed because $(k,y)\sim (0,y)$. Under ${\pi }_1(L_y)\cong K_y=k\mathbb{Z}$ it corresponds to the positive generator $k$. Its forward holonomy is the germ of $f^{-k}$ by [F1]. Its representation value is the inverse germ, that of $f^k$; these two germs generate the same cyclic group, proving claim 2.

The identity and finite-order cases. If $f=\mathrm{i}\mathrm{d}$, the quotient is the torus with its product foliation, every $K_y=\mathbb{Z}$, and its leaf-loop holonomy germs are identities. For a rotation by $2\pi p\mathrm{/}q$, with $p\in \mathbb{Z}$ and $q\ge 1$, put $d=q\mathrm{/}\gcd (p,q)$. The rotation has exact order $d$, and $f^k(y)=y$ exactly when $d$ divides $k$, so $K_y=d\mathbb{Z}$ for every $y$. Thus every leaf is a circle and every leaf-loop holonomy germ is the identity, since $f^k=\mathrm{i}\mathrm{d}$ for $k\in d\mathbb{Z}$. A path over one base circuit need not be closed and may have the nonidentity germ of $f^{-1}$; claim 3 concerns loops in the leaves.

Conclusion. Claims 1, 2 and 3 are established in steps 1.1, 1.2 and 1.3: the leaves of $F_f$ are circles exactly for periodic orbits, the holonomy of a periodic leaf is generated by $f^{-k}$, and for the identity or a finite-order rotation all holonomy is trivial even though the leaves are circles.