An asymptotic behavior of the dilatation for a family of pseudo-Anosov braids

TL;DR

This paper investigates the asymptotic behavior of dilatation in a specific family of pseudo-Anosov braids, providing formulas, monotonicity, and examples with extreme properties.

Contribution

It introduces an inductive formula for dilatation, analyzes its asymptotic behavior, and constructs examples with minimal dilatation and large volume.

Findings

Derived an inductive formula for dilatation

Established monotonicity of dilatation in the family

Constructed examples with arbitrarily small dilatation and large volume

Abstract

The dilatation of a pseudo-Anosov braid is a conjugacy invariant. In this paper, we study the dilatation of a special family of pseudo-Anosov braids. We prove an inductive formula to compute their dilatation, a monotonicity and an asymptotic behavior of the dilatation for this family of braids. We also give an example of a family of pseudo-Anosov braids with arbitrarily small dilatation such that the mapping torus obtained from such braid has 2 cusps and has an arbitrarily large volume.