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

Abstract

We consider homeomorphisms ƒ of a punctured 2-disk D2 \ P, where P is a finite set of interior points of D2, which leave the boundary points fixed. Any such homeomorphism induces an automorphism ƒ* of the fundamental group of D2 \ P. Moreover, to each homeomorphism ƒ, a matrix Bƒ (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 ƒ. First, we consider the lower bound for the entropy found by R. Bowen by using the growth rate of the induced automorphism ƒ*. 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 Bƒ (t), where t ∈ ℂ, and slightly improve his result.