#### Volume 6, issue 2 (2006)

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A family of pseudo-Anosov braids with small dilatation

### Eriko Hironaka and Eiko Kin

Algebraic & Geometric Topology 6 (2006) 699–738
 arXiv: 0904.0594
##### Abstract

This paper describes a family of pseudo-Anosov braids with small dilatation. The smallest dilatations occurring for braids with 3,4 and 5 strands appear in this family. A pseudo-Anosov braid with $2g+1$ strands determines a hyperelliptic mapping class with the same dilatation on a genus–$g$ surface. Penner showed that logarithms of least dilatations of pseudo-Anosov maps on a genus–$g$ surface grow asymptotically with the genus like $1∕g$, and gave explicit examples of mapping classes with dilatations bounded above by $log11∕g$. Bauer later improved this bound to $log6∕g$. The braids in this paper give rise to mapping classes with dilatations bounded above by $log\left(2+\sqrt{3}\right)∕g$. They show that least dilatations for hyperelliptic mapping classes have the same asymptotic behavior as for general mapping classes on genus–$g$ surfaces.

##### Keywords
pseudo-Anosov, braid, train track, dilatation, Salem–Boyd sequences, fibered links, Smale horseshoe map
##### Mathematical Subject Classification 2000
Primary: 37E30, 57M50
##### Publication
Received: 23 July 2005
Revised: 13 April 2006
Accepted: 26 April 2006
Published: 12 June 2006
##### Authors
 Eriko Hironaka Department of Mathematics Florida State University Tallahassee FL 32306-4510 USA Eiko Kin Department of Mathematical and Computing Sciences Tokyo Institute of Technology 2-12-1-W8-45 Oh-okayama Meguro-ku Tokyo 152-8552 Japan