FPM 2005, vol. 11, no. 5, pp. 47-55

FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2005, VOLUME 11, NUMBER 5, PAGES 47-55

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

O. N. Biryukov

Abstract

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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 $t$ Î C, and slightly improve his result.

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