#### Volume 7, issue 3 (2007)

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The Burau estimate for the entropy of a braid

### Gavin Band and Philip Boyland

Algebraic & Geometric Topology 7 (2007) 1345–1378
 arXiv: arxiv:math.DS/0612716
##### Abstract

The topological entropy of a braid is the infimum of the entropies of all homeomorphisms of the disk which have a finite invariant set represented by the braid. When the isotopy class represented by the braid is pseudo-Anosov or is reducible with a pseudo-Anosov component, this entropy is positive. Fried and Kolev proved that the entropy is bounded below by the logarithm of the spectral radius of the braid’s Burau matrix, $B\left(t\right)$, after substituting a complex number of modulus $1$ in place of $t$. In this paper we show that for a pseudo-Anosov braid the estimate is sharp for the substitution of a root of unity if and only if it is sharp for $t=-1$. Further, this happens if and only if the invariant foliations of the pseudo-Anosov map have odd order singularities at the strings of the braid and all interior singularities have even order. An analogous theorem for reducible braids is also proved.

##### Keywords
Dynamical systems, Braid group, Burau representation
##### Mathematical Subject Classification 2000
Primary: 37E30
Secondary: 37B40, 20F36, 20F29