We study semigroups of composition operators acting on the Besov spaces \(\mathcal {B}_p\) , where they exhibit some new behaviors relative to many classical spaces. Often for a Banach space X of analytic functions on the unit disk, the maximal closed space of strong continuity, \([\varphi _t, X]\) , exists for every semigroup \(\{\varphi _t\}\) of analytic self-maps of the disk, and the question whether \([\varphi _t, X]\) equals X itself has an answer independent of \(\{\varphi _t\}\) . Such is the case for the Hardy and Bergman spaces, Bloch, BMOA, and \(H^{\infty }\) . For the disk algebra A, \([\varphi _t, A] = A\) precisely when \(\{\varphi _t\} \subset A\) . For \(\mathcal {B}_p\) with \(p \ge 2\) , every \(\{\varphi _t\} \subset \mathcal {B}^p\) and always \([\varphi _t, \mathcal {B}_p] = \mathcal {B}_p\) , but this fails when \(1< p < 2\) . We give an example where \(\{\varphi _t\} \subset \mathcal {B}_p\) and yet the induced composition operators \(\{C_t\}\) are not bounded on \(\mathcal {B}_p\) and we do not know if \([\varphi _t,\mathcal {B}_p]\) exists. If it does exist, it cannot be equal to \(\mathcal {B}_p\) . Under the hypothesis that there is a uniform bound for the operator norms of the \(\{C_t\}\) , \(0 \le t \le 1\) , we characterize the semigroups \(\{\varphi _t\}\) such that \([\varphi _t, \mathcal {B}_p] = \mathcal {B}_p\) .
