Boundedness of bundle diffeomorphism groups over a circle

In this paper we study boundedness of bundle diffeomorphism groups over a circle. For a fiber bundle $\pi : M \to S^{1}$ with fiber $N$ and structure group $\varGamma$ and $r \in \mathbb{Z}_{\geq 0} \cup \{ \infty \}$ we distinguish an integer $k = k(\pi, r) \in \mathbb{Z}_{\geq 0}$ and construct a function $\nu^{\wedge} : \mathrm{Diff}^{r}_{\pi}(M)_{0} \to \mathbb{R}_{k}$. When $k \geq 1$, it is shown that the bundle diffeomorphism group $\mathrm{Diff}^{r}_{\pi}(M)_{0}$ is bounded and $clb_{\pi}d\, \mathrm{Diff}^{r}_{\pi}(M)_{0} \leq k + 3$, if $\mathrm{Diff}^{r}_{\varrho,c}(E)_{0}$ is perfect for the trivial fiber bundle $\varrho : E \to \mathbb{R}$ with fiber $N$ and structure group $\varGamma$. On the other hand, when $k = 0$, it is shown that $\nu^{\wedge}$ is a unbounded quasimorphism, so that $\mathrm{Diff}^{r}_{\pi}(M)_{0}$ is unbounded and not uniformly perfect. We also describe the integer $k$ in term of the attaching map $\varphi$ for a mapping torus $\pi : M_{\varphi} \to S^{1}$ and give some explicit examples of (un)bounded groups.