Quadratic-time computations for pseudo-Anosov mapping classes

TL;DR

This paper presents a quadratic-time algorithm for computing the stretch factor and invariant measured foliations of pseudo-Anosov mapping classes, improving computational efficiency in this area.

Contribution

It introduces the first known quadratic-time algorithm for computing stretch factors and measured foliations of pseudo-Anosov elements in the mapping class group.

Findings

Algorithm runs in quadratic time relative to input length

Successfully computes stretch factors as largest eigenvalues

Outputs include train tracks and eigenvectors for foliations

Abstract

We give a quadratic-time algorithm to compute the stretch factor and the invariant measured foliations for a pseudo-Anosov element of the mapping class group. As input, the algorithm accepts a word (in any given finite generating set for the mapping class group) representing a pseudo-Anosov mapping class, and the length of the word is our measure of complexity for the input. The output is a train track and an integer matrix where the stretch factor is the largest real eigenvalue and the unstable foliation is given by the corresponding eigenvector. This is the first algorithm to compute stretch factors and measured foliations that is known to terminate in sub-exponential time.