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Fundamentalnaya i Prikladnaya Matematika, 2005, Volume 11, Issue 5, Pages 47–55
(Mi fpm862)
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This article is cited in 1 scientific paper (total in 1 paper)
A bound for the topological entropy of homeomorphisms of a punctured two-dimensional disk
O. N. Biryukov Kolomna State Pedagogical Institute
Abstract:
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.
Citation:
O. N. Biryukov, “A bound for the topological entropy of homeomorphisms of a punctured two-dimensional disk”, Fundam. Prikl. Mat., 11:5 (2005), 47–55; J. Math. Sci., 146:1 (2007), 5483–5489
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https://www.mathnet.ru/eng/fpm862 https://www.mathnet.ru/eng/fpm/v11/i5/p47
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Abstract page: | 306 | Full-text PDF : | 130 | References: | 46 |
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