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
$D2\backslash P$, where
$P$ is
a finite set of interior points of $D2$, which leave
the boundary points fixed.
Any such homeomorphism induces an automorphism $f$_{*} of the
fundamental group of $D2\backslash P$.
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.

Location: http://mech.math.msu.su/~fpm/eng/k05/k055/k05504h.htm

Last modified: February 26, 2006