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 $\mathrm{\Gamma }$ and $r\in {\mathbb{Z}}_{\ge 0}\cup \{\mathrm{\infty }\}$ we distinguish an integer $k=k(\pi ,r)\in {\mathbb{Z}}_{\ge 0}$ and construct a function $\widehat{\nu }:{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\pi }(M{)}_0\to {\mathbb{R}}_k$. When $k\ge 1$, it is shown that the bundle diffeomorphism group ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\pi }(M{)}_0$ is uniformly perfect and $cl{b}_{\pi }{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\pi }^r(M{)}_0\le k+3$, if ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\rho ,c}(E{)}_0$ is perfect for the trivial fiber bundle $\rho :E\to \mathbb{R}$ with fiber $N$ and structure group $\mathrm{\Gamma }$. On the other hand, when $k=0$, it is shown that $\widehat{\nu }$ is a unbounded quasimorphism, so that ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\pi }(M{)}_0$ is unbounded and not uniformly perfect. We also describe the integer $k$ in term of the attaching map $\phi$ for a mapping torus $\pi :M_{\phi }\to S^1$ and give some explicit examples of (un)bounded groups.