Semigroups of composition operators on the Besov spaces

In this paper, we characterize the strong continuity of composition semigroups on analytic Besov spaces \(B_{p}(1<p<\infty ).\) First, we show that every semigroup of composition operators \(\{C_{\varphi _{t}}\}\) are strongly continuous on \(B_{p}(2\le p<\infty ).\) However, we can find a semigroup \(\{\varphi _t\}\) such that the induced composition operator \(C_{\varphi _t}\) is not even bounded on \(B_p(1<p<2).\) We contribute novel counterexamples grounded in the geometric properties of the image domain of Kœnigs function to illustrate this point. Moreover, we provide a sufficient condition ensuring the strong continuity of any semigroup of composition operators in \(B_{p}(1<p<\infty ).\) Additionally, we establish that \(\{C_{\varphi _{t}}\}\) is not uniformly continuous on \(B_{p}(1<p<\infty ),\) unless \(\{\varphi _{t}\}\) is trivial.