A bound for the topological entropy of homeomorphisms of a punctured two-dimensional disk

We consider homeomorphisms $f$ of a punctured 2-disk $D^2\setminus P$, where $P$ is a finite set of interior points of $D^2$, which leave the boundary points fixed. Any such homeomorphism induces an automorphism $f_*$ of the fundamental group of $D^2\setminus P$. Moreover, to each homeomorphism $f$, a matrix $B_f(t)$ from $\mathrm{GL}(n,\mathbb Z[t,t^{-1}])$ can be assigned by using the well-known Burau representation. The purpose of this paper is to find a nontrivial lower bound for the topological entropy of $f$. First, we consider the lower bound for the entropy found by R. Bowen by using the growth rate of the induced automorphism $f_*$. Then we analyze the argument of B. Kolev, who obtained a lower bound for the topological entropy by using the spectral radius of the matrix $B_f(t)$, where $t\in\mathbb C$, and slightly improve his result.