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 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 P. Moreover, to each homeomorphism f, a matrix B f (t) from GL(n, 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 ℂ, and slightly improve his result. © 2007 Springer Science+Business Media, Inc.